


































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































\ 






X? 2 l- V* 














HANDBOOK 

OF 

MILITARY INFRARED TECHNOLOGY 


Editor 

William L. Wolfe 

The University of Michigan 


1965 


Office of Naval Research 
Department of the Navy 
Washington, D.C. 


For sale by the Superintendent of Document?, U.S. Government Printing Office 

Washington, D.C. 20402 - Price $3.75 





Handbook of Military Infrared Technology 


* * * 


Compiled by The University of Michigan under 
Contract Nonr 1224 (12) with the Office of Naval 
Research and ARP A Order 161 with the Advanced 
Research Projects Agency. Published by the Naval 
Research Laboratory and printed by the U.S. 
Government Printing Office. Reproduction is per¬ 
mitted for any purposes of the U.S. Government. 

Library of Congress Catalog Card No: 65-62266 



- 196 / 







Contributors 


James L. Alward, Research Associate, The University of Michigan, Institute of Science 
and Technology, Ann Arbor, Michigan. Spatial Frequency Filtering 

Richard Barakat, Research Laboratory, Itek Corporation, Lexington, Massachusetts. 
Optical Frequency-Response Techniques 

C. E. Dunning, Deputy Manager, Research and Development, Astrionics Division, 
Aerojet-General Corporation, Azusa, California. Targets 

R. J. Hembach, Head, Thermal Testing, Grumman Aircraft Engineering Corporation, 
Bethpage, New York. Spacecraft Thermal Design 

L. H. Hemmerdinger, Group Leader, O.A.O. Thermal Control, Grumman Aircraft 
Engineering Corporation, Bethpage, New York. Spacecraft Thermal Design 

J. A. Jenney, Graduate Research Assistant, The University of Michigan, Institute 
of Science and Technology, Ann Arbor, Michigan. Physical Constants and Conver¬ 
sion Factors 

Richard Kauth, Research Associate, The University of Michigan, Institute of Science 
and Technology, Ann Arbor, Michigan. Backgrounds 

Thomas Limperis, Associate Research Physicist, The University of Michigan, Institute 
of Science and Technology, Ann Arbor, Michigan. Detectors 

Lawrence D. Lorah, Senior Scientist, Mithras, Inc., Cambridge, Massachusetts. Aero¬ 
dynamic Influences on Infrared System Design 

K. Robert Morris, Graduate Research Assistant, The University of Michigan, Institute 
of Science and Technology, Ann Arbor, Michigan, (now with Conduction, Inc., Ann 
Arbor, Michigan). Control Systems 

Fred E. Nicodemus, Senior Scientist, Sylvania Electric Systems, Electronic Defense 
Laboratories, Mountain View, California. Radiation Theory; Targets 

Richard A. Phillips, Assistant in Research, The University of Michigan, Institute of 
Science and Technology, Ann Arbor, Michigan. Physical Constants and Conversion 
Factors 

Gilbert N. Plass, Professor of Atmospheric and Space Sciences, Southwest Center for 
Advanced Studies, Dallas, Texas. Atmospheric Phenomena 

Eugene S. Rubin, President, Mithras, Inc., Cambridge, Massachusetts. Aerodynamic 
Influences on Infrared System Design 

Sol Shapiro, Department Head, General Precision, Inc., Wayne, New Jersey. System 
Design 


ill 


David Silvermetz, Staff Engineer, Servo Corporation of America, Hicksville, New York. 
Preamplifiers and Associated Circuits 

Warren J. Smith, Vice President — Research, Optical Systems Division, Infrared In¬ 
dustries, Inc., Santa Barbara, California. Optics; Optical Systems 

Allan L. Sorem, Research Associate, Research Laboratories, Eastman Kodak Company, 
Rochester, New York. Film 

Gwynn H. Suits, Head, The University of Michigan, Institute of Science and Technology, 
Ann Arbor, Michigan. Film 

William L. Wolfe, Research Engineer, The University of Michigan, Institute of Science 
and Technology, Ann Arbor, Michigan 

Arthur E. Woodward, Engineering Manager, Servo Corporation of America, Hicksville, 
New York. Preamplifiers and Associated Circuits 

Harold Yates, Chief Engineer, Field Research and Systems Department, Barnes Engi¬ 
neering Company, Stamford, Connecticut. Atmospheric Phenomena 


IV 


Preface 


Origins 

Infrared techniques, particularly as applied to military problems, have been 
developing in a gradual way for many, many years. One of the first main 
uses was in World War II in such applications as the DOVE guided bomb and 
the German Lichtsprecher. Since then, the Sidewinders, the Midas satellites, 
communication devices, and a myriad of other instruments and systems have 
been proposed, designed, developed, and used. Whereas in the early 1950’s 
a small group of researchers could convene and discuss most of the infrared 
problems that were then facing the nation, now such meetings encompass 
between five hundred and a thousand people. The techniques have changed, 
the components have improved, and the means of use for these components 
have become more sophisticated and more complicated. 

Early in 1961, A1 Canada, who was then at ARPA —the Advanced Research 
Projects Agency — and the editor of this Handbook had a series of conversations 
including discussions of ways to improve communications among infrared 
workers and of methods by which the talents of these workers could be used 
more effectively. Both of us were aware of the capabilities of organizations 
like IRIS (the Infrared Information Symposia) and IRIA (the Infrared Informa¬ 
tion and Analysis Center), but we felt that something different would be 
useful. 

In the field of microwaves and radar, one of the first unclassified treatments 
of their application to fairly complicated guidance and control and related 
problems was a series of books written by staff members of the Radiation 
Laboratory of Massachusetts Institute of Technology. Many of these became 
classics. We felt that the infrared field could benefit from a similar publica¬ 
tion program. Unfortunately, monetary problems forced us to modify our 
first hopes drastically. The result is this Handbook. 

ARPA, via the Office of Naval Research, contracted with the University of 
Michigan to supervise writing of this book, and ONR requested the Naval 
Research Laboratory to arrange for its publication. ARPA suggested that a 
technical writing firm be retained for assistance, and the University, after 
evaluating competitive bids, selected the McGraw-Hill Book Company, 
Training Materials and Information Services Division —now Information and 
Training Services Division of the F. W. Dodge Company, a Division of 
McGraw-Hill, Inc. This group, whose responsibility was to the University, 
selected the authors for the several chapters and sections; they were in turn 
responsible to this McGraw-Hill division. 


v 


The project was initiated in August of 1962. Much manuscript was pre¬ 
pared during the ensuing year. In June of 1963 The University of Michigan 
elected to complete the final editing and writing without the assistance of 
McGraw-Hill. Some chapters were then finally prepared, and all chapters 
were revised somewhat, edited, and copy edited. The first portion of the 
manuscript was sent to the Naval Research Laboratory on February 25 and 
the last in August of 1964. 

Credits 

Hundreds of people have helped. The people whose help should be ac¬ 
knowledged in this preface are those who have made the most substantial 
contributions to the Handbook, although this should in no way indicate that 
the less substantial contributions were not valuable. The list of contributors 
(page iii) gives the chapter authors with their present affiliations, and the 
titles of their chapters. 

People at The University of Michigan who should receive particular credit 
for their endeavors on this Handbook are Mrs. Hilda Taft and the IRIA clerical 
staff. Mrs. Taft painstakingly copy edited virtually the entire manuscript 
and it was she who brought to my attention all the idiosyncracies, irregu¬ 
larities, and inconsistencies in symbols and equations. If they are incorrect 
it is because I failed to take proper action after they were brought to my 
attention. The IRIA clerical staff, Dorothy Curtis, Beatrice Godin, Myrtle 
Kreie, Beth Larson, Marie Nichols, and Sonya Kennedy kept track of all the 
correspondence and the many, many revisions and different versions of each 
of the chapters, as well as the stray figures and reference checking during this 
work. I am also indebted to Dr. George J. Zissis, who critically read a great 
deal of the manuscript and provided useful comments about many chapters 
but particularly about Chapters 2, 3, and 4. J. P. Livisay and John Duncan 
criticized Chapter 6, and John Duncan wrote one section of Chapter 6; Donald 
M. Szeles provided some additional material for Chapters 9 and 10; Roy J. 
Nichols criticized Chapter 21; John Gebhardt calculated considerable material 
for some of the tables in Chapter 22. 

We received criticisms and comments from many workers not associated 
with The University of Michigan. Lucien M. Biberman, Institute for Defense 
Analyses, Washington, D.C., and Dr. E. D. McAlister, Director, Applied 
Oceanography Group, University of California, Scripps Institution of Ocean¬ 
ography, San Diego, California, provided criticisms for Chapter 5; Dr. Stanley 
S. Ballard, University of Florida, Department of Physics, Gainesville, Florida, 
provided useful comments about Chapters 7 and 8; Dr. R. Clark Jones, 
Polaroid Corporation, Cambridge, Massachusetts, Dr. Henry Levinstein, 
Physics Department, Syracuse University, Syracuse, New York, and W. L. 
Eisenman, Naval Ordnance Laboratory, Corona, California, criticized the 
tables of Chapter 11; Dr. J. Stanley Buller, Santa Barbara Research Center, 
Goleta, California, provided useful criticisms for Chapter 12; K. L. DeBrosse, 


vi 


ITT Industrial Laboratories, Fort Wayne, Indiana, commented on Chapter 14; 
and T. P. Jackson, Aerojet-General Corporation, Azusa, California, provided 
useful criticisms for Chapter 20. 

All these people have helped by their comments, criticisms, or creative 
writing. It is only fair and truthful to add that in the end I have taken it 
upon myself to make some changes either as a result of what was written, as 
a result of the criticisms, or in spite of both. I hope that my changes have 
improved the Handbook. If you find that you argue with or take exception to 
either what is in this book or what is not, you are arguing principally with my 
decisions. 

The following individuals and companies provided information that was 
useful in preparing this Handbook: 

J. G. Sample, Raytheon Co. Richard Rossi, Malakar Laboratories 


L. D. McGlauchlin, Minneapolis-Honeywell 
Regulator Co. 

R. W. Weidmer, Westinghouse Electric 
Corp. 

J. R. Kittler and P. L. Rice, Garrett Corp. 
Allen Olsen, Naval Ordnance Test Station 
Frank Kocsis, Barnes Engineering Co. 
Joseph Jerger, Servo Corporation of 
America 

Wayne McKusick, Eastman Kodak Co. 

G. C. Higgins, Eastman Kodak Co. 

John Howard, Air Force Cambridge 
Research Laboratories 


Ann Arbor, Michigan 
April 1965 


J. Harkness, Librascope Division, General 
Precision Co. 

John Harvell, A. D. Little Co. 

L. L. Reynolds, The RAND Corp. 

George R. Pruett, Texas Instruments 
Incorporated 

Stanley Wallack, Leesona Moos Corp. 
Radiation Electronics, Inc., Infrared 
Detector Department 
E. M. Scott, Engelhard Industries, Inc. 
Ralph Stair, National Bureau of 
Standards 

Fish Schurman Corp. 


William L. Wolfe 


Vll 




















































































Contents 


Chapter Page 

Contributors . iii 

Preface . v 

1. INTRODUCTION . 1 

William L. Wolfe, The University of Michigan 

2. RADIATION THEORY. 3 

William L. Wolfe, The University of Michigan 

Fred E. Nicodemus, Sylvania Electronic Systems 

3. ARTIFICIAL SOURCES. 31 

4. TARGETS. 57 

C. E. Dunning, Aerojet-General Corporation 

Fred E. Nicodemus, Sylvania Electronic Systems 

5. BACKGROUNDS. 95 

Richard Kauth, The University of Michigan 

6. ATMOSPHERIC PHENOMENA.175 

Gilbert N. Plass, Southwest Center for Advanced Studies 

Harold Yates, Barnes Engineering Company 

7. OPTICAL COMPONENTS.281 

William L. Wolfe, The University of Michigan 

8. OPTICAL MATERIALS.315 

William L. Wolfe, The University of Michigan 

9. OPTICS.369 

Warren J. Smith, Infrared Industries, Inc. 

10. OPTICAL SYSTEMS. ' .421 

Warren J. Smith, Infrared Industries, Inc. 

11. DETECTORS.457 

T. Limperis, The University of Michigan 


IX 















12. DETECTOR COOLING SYSTEMS.519 

13. FILM.569 

Allan L. Sorem, Eastman Kodak Company 

Gwynn H. Suits, The University of Michigan 

14. PREAMPLIFIERS AND ASSOCIATED CIRCUITS.583 

Arthur E. Woodward and David Silvermetz, 

Servo Corporation of America 

15. OPTICAL FREQUENCY-RESPONSE TECHNIQUES.613 

R. Barakat, Itek Corporation 

16. SPATIAL FREQUENCY FILTERING.645 

James Alward, The University of Michigan 

17. CONTROL SYSTEMS.661 

K. R. Morris, The University of Michigan 

18. SYSTEM DESIGN.729 

Sol Shapiro, General Precision, Inc. 

William L. Wolfe, The University of Michigan 

19. INFRARED MEASURING INSTRUMENTS.757 

20. SPACECRAFT THERMAL DESIGN.783 

L. H. Hemmerdinger and R. J. Hembach, 

Grumman Aircraft Engineering Corporation 

21. AERODYNAMIC INFLUENCES ON 

INFRARED SYSTEM DESIGN.825 

Lawrence D. Lorah and Eugene Rubin, Mithras, Inc. 

22. PHYSICAL CONSTANTS AND 

CONVERSION FACTORS.853 

J. A. Jenney and Richard Phillips, 

The University of Michigan 

SUBJECT INDEX.883 


x 














Special Figure, Table, and 
Text Acknowledgments 


The parts of the Handbook itemized below are printed with the permissions indicated. 
Chapter 2 

Fig. 2-1: General Electric Company 
Fig. 2-2: Block Associates, Inc. 

Fig. 2-3: Admiralty Research Laboratory (Ministry of Defence, U.K.) 

Chapter 3 

Fig. 3-1: Revue D’Optique 

Quotation in Section 3.1.3: American Society of Mechanical Engineers 
Quotation in Section 3.2; Fig. 3-3, Fig. 3-4, and Fig. 3-5: National Bureau of Standards 
Fig. 3-6: Review of Scientific Instruments 

Chapter 4 

Fig. 4-3 and 4-6: John Wiley & Sons, Inc. 

Fig. 4-7: General Dynamics/Convair 

Fig. 4-8 and 4-9: Space/Aeronautics 

Fig. 4-11 and 4-12: Harvard University Press 

Chapter 5 

Fig. 5-1 to 5-9 incl.: Optical Society of America 
Fig. 5-11, 5-12, and 5-18: Pergamon Press, Inc. 

Fig. 5-13: Academic Press, Inc. 

Fig. 5-14 and 5-15: The Territorial Magazine 
Fig. 5-16 and 5-20: John Wiley & Sons, Inc. 

Fig. 5-17: Clarendon Press, Oxford 
Fig. 5-19: Nature 

Fig. 5-21 and 5-22, also Tables 5-2 to 5-7 incl.: Aerojet-General Corporation 
Fig. 5-23, 5-24, 5-25, also Table 5-8: The RAND Corporation 
Fig. 5-33 to 5-38 incl.: McGraw-Hill, Inc. 

Fig. 5-43 to 5-58 incl.: Air Weather Service (MATS) 

Fig. 5-61 to 5-68 incl., 5-92, and 5-97: Optical Societyof America 
Fig. 5-93 and 5-94: Institute of Geophysics, University of California 
Fig. 5-95 and 5-96: Science 

Chapter 7 

Fig. 7-1 to 7-8 incl.: Servo Corporation of America 
Fig. 7-9 and 7-10: Barnes Engineering Co. 

Fig. 7-12 and 7-13: U.S. Army Frankford Arsenal 
Fig. 7-15 and 7-27: Optical Coating Laboratory, Inc. 

Fig. 7-18: Barnes Engineering Company; The Physical Review 
Fig. 7-19, 7-20, and 7-21: A. Smakula 
Fig. 7-22 and 7-23: W. H. Freeman and Co. 

Fig. 7-24: Bausch & Lomb, Inc. 


XI 


Chapter 7 (continued) 

Fig. 7-25: Eastman Kodak Co. 

Fig. 7-26: Infrared Industries, Inc. 

Fig. 7-28: Optical Society of America 
Fig. 7-31, 7-32, and 7-34: McGraw-Hill, Inc. 

Fig. 7-35, 7-36, and 7-37: Reinhold Publishing Corp. 

Chapter 8 

Fig. 8-19, 8-31, and 8-32, also Table 8-21: Optical Society of America 
Fig. 8-21, 8-23 to 8-30 inch, 8-33, 8-34, and 8-35: Bausch & Lomb, Inc. 

Fig. 8-22: Union Carbide Corp., Stellite Division 
Fig. 8-37: Purdue Research Foundation 

Fig. 8-39: Infrared Division, Research Dept., U.S. Naval Ordnance Laboratory, 
Corona, Calif. 

Chapter 11 

Section 11.4: Syracuse University and U.S. Naval Ordnance Laboratory, 

Corona, Calif. 

Chapter 12 

Table 12-1: Cryogenic Engineering Co. 

Fig. 12-5: Barnes Engineering Co. 

Fig. 12-7: AiResearch Manufacturing Co. 

Fig. 12-9: Santa Barbara Research Center 
Fig. 12-10: ITT Industrial Laboratories Division 
Fig. 12-13: AiResearch Manufacturing Co. 

Fig. 12-19: John Wiley & Sons, Inc. 

Fig. 12-22 to 12-25 incl.: Borg-Wamer Thermoelectrics 

Chapter 13 

Fig. 13-1 to 13-18 incl.: Eastman Kodak Co. 

Chapter 1 7 

Fig. 17-9 and 17-12, also Tables 17-1 and 17-2: McGraw-Hill, Inc. 

Chapter 19 

Fig. 19-10: Unicam Instruments Ltd. 

Fig. 19-11, 19-12, and 19-16: Perkin-Elmer Corp. 

Fig. 19-13: Barnes Engineering Co. 

Fig. 19-14: Block Engineering, Inc. 

Fig. 19-15: Beckman Instruments, Inc. 

Fig. 19-17 and 19-19: W. H. Freeman & Co. 

Fig. 19-20: Reinhold Publishing Corp. 

Fig. 19-21: The Institute of Physics and The Physical Society, U.K. 

Fig. 19-22: W. H. Freeman & Co. 

Chapter 20 

Fig. 20-1, 20-2, and 20-3: American Institute of Aeronautics and Astronautics 
Fig. 20-4, 20-5, and 20-6: McGraw-Hill, Inc. 

Fig. 20-7: American Meteorological Society 
Table 20-3: John Wiley & Sons, Inc. 

Fig. 20-12: American Institute of Aeronautics and Astronautics 

Fig. 20-14 and 20-19: D. K. Edwards, K. E. Nelson, R. D. Roddick, and J. T. Gier 


Xll 


Chapter 21 

Fig. 21-11 and 21-12: American Institute of Aeronautics and Astronautics 
Chapter 22 

Fig. 22-1, 22-4, and 22-5: Aerojet-General Corp. 

Table 22-1: American Institute of Physics 

Tables 22-6, 22-7, 22-8, 22-12, 22-14, and 22-18: Chemical Rubber Co. and 
John Wiley & Sons, Inc. 

Table 22-9: University of Chicago Press 

Tables 22-15 and 22-17: Chemical Rubber Co. and General Electric Co. 


xm 


Handbook of Military Infrared Technology 


Chapter 1 

INTRODUCTION 

William L. Wolfe 

The University of Michigan 


The chapters of this Handbook are arranged in a sequence that is now almost tradi¬ 
tional, and it is logical. The radiators come first, then the medium of propagation, the 
receiver system, the transducers and electronics, and finally a number of special appli¬ 
cations. Thus Chapters 2, 3, 4, and 5 deal with basic radiation laws, blackbody simula¬ 
tors, and the properties of targets and of backgrounds. Not very much attention is 
paid to the more difficult problem of calculating the amount and kind of gaseous radia¬ 
tion because it is not generally a problem for the systems engineer, whereas envelope 
calculations based on slide rules and formulas most certainly are. Traceability of 
instrument performance to the National Bureau of Standards is more and more a real 
question; therefore the entire problem of radiometrics has been dealt with in more 
detail than is usual for a handbook. 

Chapter 6 deals with atmospheric absorption and contains some material on scat¬ 
tering and scintillation. Absorption processes and the calculation of absorption are 
relatively well known, and a detailed explanation of the theory and methods of calcula¬ 
tion are given. The chief problem here is knowing the atmospheric composition. 
Much must still be done concerning the loss due to scattering, and with scintillation; 
here the terms are not even well defined. 

The next group of chapters deals with optics and optical design. Considerable detail 
is given on design techniques because so little is available elsewhere. The basic 
nomenclature of Conrady is followed. Components and materials are discussed. A 
condensation of material contained in an IRIA* state-of-the-art report is given, with 
an augmentation on glasses. Although many optical components are bought by 
specification and fabrication to order, some and even some lenses do exist "on the shelf.” 
As many as possible of these commercially available optical components are listed. 

The chapter on detectors is relatively short, but the design engineer should find the 
extensive table of considerable use. For the first time a readily available useful display 
of most detector concepts appears in print. Methods of test vary, but those listed here 
have some measure of acceptance. 

Detectors are often cooled, of course; Chapter 12 presents the most comprehensive 
table of coolers for infrared detectors ever published (more have probably come on the 
market since this book went to press). In addition, Stirling cycle systems and tech¬ 
niques using samples of solid hydrogen, helium, and the like are being developed, but 
they are not sufficiently well along for inclusion. 

*Infrared Information and Analysis Center, a part of the Infrared Laboratories of the Institute of 
Science and Technology at The University of Michigan, Ann Arbor, Michigan. The report referred 
to is 2389-11-S, January 1959. 


1 





2 


INTRODUCTION 


The chapters on film and preamplifiers conclude the treatment of the usual "parts” 
of an infrared system. Although there are many brochures on film and its performance, 
none has ever appeared that was couched in radiometric terms. It is apparently true 
that all infrared film is manufactured by the Eastman Kodak Company. 

The remaining chapters deal with certain special features of infrared engineering. 
In this connection, the method of writing on systems design is of particular note. It 
should be self-evident that no two engineers do system design in the same fashion. 
Some general approaches have been discussed by such texts as Goode and Machol;* 
these usually include a respectable amount of probability mathematics and concepts, 
game theory, and block-diagram operations or signal-flow graphs. They also describe 
certain common-sense approaches to the simplification of the problem by dissection. 
To a large extent Chapter 18 is just such a description for infrared systems. 

The authors hope that this Handbook will be used like most other handbooks. Oc¬ 
casionally the user will browse for ideas. More often he will be searching for the answer 
to a specific problem —the necessary data or the required formulas or techniques. 
The index should be the key to the answer for this need. It has been laboriously pre¬ 
pared to include references, cross references, and other helpful clues. The organization 
should serve those who browse and should help in specific searches by having, near the 
referenced, searched-for item, others that are closely associated with it. 

This Handbook is not a state-of-the-art report on all phases of infrared systems and 
components; it is not and was not meant to be. But in some senses it has to be. A 
handbook is usually a source of useful information —data, equations, concepts, and 
techniques. It includes those things that are useful for undertaking certain develop¬ 
ment and research tasks, but it is not the last, up-to-the-minute word on all subjects. 

As is so for every handbook, this one is neither completely up-to-date nor entirely 
comprehensive. The field is dynamic, and the sum of all material of interest to every¬ 
one in the field is an appreciable per cent of infinity. The references provide one clue 
for obtaining more information on any given subject. Another source is the various 
information centers dealing directly with infrared topics or touching upon infrared as a 
peripheral interest. Some of these centers are listed below. The Science Information 
Exchange, Suite 313, Universal Bldg., 1825 Connecticut Avenue, N.W., Washington, 
D.C. should be consulted for further lists of Centers in existence and their topics. 
1-4. At the Institute of Science and Technology, The University of Michigan, Box 618, 
Ann Arbor, Michigan: IRIA, the Infrared Information and Analysis Center; 
BAMIRAC, the Ballistic Missile Radiation Analysis Center; TABSAC, the Target 
and Backgrounds Signature Analysis Center; BAC, the Background Analysis 
Center. 

5. SPIA-LPIA, the Solid and Liquid Propellant Information Agency of the Applied 
Physics Laboratory of The Johns Hopkins University, 8621 Georgia Avenue, 
Silver Spring, Maryland. 

6. RACIC, the Remote Areas Conflict Information Center, Battelle Memorial In¬ 
stitute, 505 King Avenue, Columbus, Ohio. 

7. CINFAC, the Counterinsurgency Information Analysis Center, at American 
University, Washington, D.C. 

8. IRIS, the Infrared Information Symposia, an organization devoted to appropriate 
timely dissemination of research and development results by meetings. Attend¬ 
ance is possible through Mr. Thomas B. Dowd, Office of Naval Research, 495 
Summer Street, Boston, Massachusetts. 

*H. Goode and R. Machol, Systems Design, Control Systems Engineering, McGraw-Hill Book 
Company, New York, 1957. 



Chapter 2 

RADIATION THEORY 


William L. Wolfe 

The University of Michigan 

Fred. E. Nicodemus 

Sylvania Electronic Systems 


CONTENTS 


2.1. Radiometric Quantities, Symbols, and Units. 4 

2.1.1. Radiometric Quantities as Field Concepts. 7 

2.1.2. Other Radiometric Quantities. 7 

2.2. Kirchhoff’s Law: Emissivity and Blackbodies. 9 

2.3. Radiation Laws. 9 

2.3.1. Planck’s Law. 9 

2.3.2. Quantum Rates in Blackbody Radiation.10 

2.3.3. Stefan-Boltzmann Law.10 

2.3.4. Rayleigh-Jeans and Wien Laws.10 

2.3.5. The Wien Displacement Law.10 

2.3.6. Maximum Difference Expression.10 

2.4. Blackbody Slide Rules.11 

2.4.1. The General Electric Rule.11 

2.4.2. The Block Rule.11 

2.4.3. The Admiralty Research Laboratory Rule.17 

2.5. Blackbody Curves.17 

2.6. Blackbody Tables.21 

2.7. Radiation Geometry.22 

2.7.1. Lambertian Sources.22 

2.8. Distributed Radiators.23 

2.9. Selective Radiators.23 

2.10. Directional Reflectance and Emissivity.23 

2.11. Summary of Equations and Constants.28 


3 



























2. Radiation Theory 


2.1. Radiometric Quantities, Symbols, and Units 

The nomenclature, symbols, and units of the most important radiometric quantities 
are listed in Table 2-1, which is based on the recommendations of the Working Group 
on Infrared Backgrounds (WGIRB) [1,2]. They include and are consistent with Amer¬ 
ican Standard Z58.1.1-1953. 


Table 2-1. Symbols, Names, and Units of Radiometric Quantities 


Symbol 

Name 

Description 


U nits 

A 

Area 

Projected area 


cm 2 

a 

Solid angle 

— 


sr 

V 

Volume 

— 


cm 3 

u 

Radiant 

energy 

— 


joule 

u 

Radiant 

energy 

density 

Radiant energy d U 
per unit volume qy 


joule cm 

P 

Radiant 

power 

Rate of transfer d U 

of radiant energy dt 


w 

W 

Radiant 

emittance 

Radiant power per unit 
area emitted from 
a surface 

dP 

dA 

w cm -2 

H 

Irradiance 

Radiant power per unit 
area incident upon 
a surface 

dP 

dA 

w cm -2 

J 

Radiant 

intensity 

Radiant power per unit 
solid angle from a 
point source 

dP 

dCi 

w sr -1 

N 

Radiance* 

Radiant power per unit 
solid angle per unit 
projected area 

d 2 p 

cos 0 dA dCt 

w sr -1 cr 

Pk 

Spectral 

radiant 

power 

Radiant power per unit 
wavelength interval 

dP 

d\ 

W fX~ x 

P v 

Spectral 

radiant 

power 

Radiant power per unit 
frequency interval 

dP 

dv 

w sec 


’Sometimes radiance is defined instead as the radiant power per unit area (not projected area) per unit solid angle 

/ d*P \ 

\N = This * s e< ? uall y correct > but it is then necessary to insert the cos d factor differently, eg., it is then 

N cos d which is invariant along a ray, and the radiance of a Lambertian surface varies with the cosine of the angle 
from the normal. 


4 





RADIOMETRIC QUANTITIES, SYMBOLS, AND UNITS 5 

Table 2-1. Symbols, Names, and Units of Radiometric Quantities (Continued) 


Symbol 

Name 

Description 


Units 

Pa 

Spectral 

Radiant power per dP 


w cm 


radiant 

unit wave da 




power 

number interval 



W x 

Spectral 

Radiant emittance per 

dW 

w cm -2 /x -1 


radiant 

unit wavelength 

dk 



emittance 

interval 



H\ 

Spectral 

Irradiance per unit 

dH 

w cm -2 g~ x 


irradiance 

wavelength interval 

dk 


Jx 

Spectral 

Radiant intensity per 

dJ 

w sr _1 p, -1 


radiant 

unit wavelength 

dk 



intensity 

interval 



N\ 

Spectral 

Radiance per unit 

dN 

w sr _1 cm -2 /x _1 


radiance 

wavelength interval 

dk 



A second set of symbols and units (Table 2-2) is as written by Penner [3] and pat¬ 
terned after Worthing. In this system, a superscript 0, e.g., R°, indicates that the quan¬ 
tity is for a blackbody, and a subscript like v or k indicates partial differentiation. 
Thus, in this terminology R\° is the spectral radiancy of a blackbody in w cm -2 /x _1 ; 
R y ° is the spectral radiancy of a blackbody in w cm -2 sec. Other investigators prefer 
to use the superscript or subscript bb or b to denote blackbody. 


Table 2-2. 
Symbol 
& 

P 


Penner Radiometric Symbols 

Definition 
Radiant energy 
Radiant energy density 




J = 


dtb 

~dt 

dSe 

an 


Power 

Radiant intensity 


R = ~ 

dA r 


W = 


aT 

dA r 


J._a d$e 

cos 6 dA r an 


Radiancy: radiant flux per 
unit area from a source 
into a hemisphere 

Radiant flux density 
(not of a source) 

Steradiancy or radiance 


/ or H or R 


a£i 

dAr 


Irradiancy 


Proposals have been made for altering the present names and symbols. The two now 
most in favor are those of R. Clark Jones and those of some NBS personnel. The latter 
would retain the words radiant emittance, radiance, irradiance, and radiant intensity. 
The system would, however, use the suffix -ance to describe measured properties of a 
particular sample and -ivity to indicate a property that is intrinsic with a material when 
used in connection with material rather than field quantities. Thus the reciprocal of 
the ratio of incident power (from a plane wave at normal incidence on a flat surface) 












6 


RADIATION THEORY 


to the reflected power perpendicular to any sample is the normal reflectance. If the 
sample has optically flat polished surfaces the quantity is reflectivity. The obvious dif¬ 
ficulty is in the words emittance and emissivity, although there are other difficulties. 

Jones proposes a more general set of specifications. He deals with geometrical 
concepts. He calls the rate of flow of any quantity from a body per unit area exitance, 
the rate impinging per unit area incidance, the rate per unit area per unit solid angle 
sterance, and the rate per unit solid angle from a point source intensity. Then appro¬ 
priate adjectival modifiers are used. Table 2-3 illustrates this system. 


Table 2-3. Jones’ Proposed Terminology* 


Geometric 

Radiometric 

Radiometric 

Quantity 

Quantity 

Quantity 

(Jones) 

(Jones) 

(Standard) 

Emittance 

Radiant exitance 

Radiant emittance 

Incidance 

Radiant incidance 

Irradiance 

Sterance 

Radiant sterance 

Radiance 

Intensity 

Radiant intensity 

Radiant intensity 

*R. C. Jones, 'Terminology in Photometry and Radiometry,” J. Opt. Soc. Am., 53, 

11, 1314 (November 1963). 


Another set of symbols (Table 2-4) is used 

in the thermal control industry and is 

gaining favor in 

a segment of the aerospace 

industries. It is also patterned after 

Worthing [4]. The systems are compared in Table 2-5. 


Table 2-4. Aerospace Radiometric Symbols 

Symbol 

Term 

Definition 

R 

Radiancy 

Power emitted per unit source 



area to a hemisphere 

— 

Steradiancy 

Power emitted per unit area 
per unit solid angle. 


Table 2-5. Comparison of Systems of Terms* 

WGIRB 

Penner 

Aerospace Jones 

Energy, U 

Energy, & 

Energy, Q - 

Energy density, u 

Energy density, p 

— — 

Power, P 

Emitted power, 

Heat rate, q — 

Radiant 

Radiancy, R 

Radiancy, R Radiant 

emittance, W 


emittance 

Irradiance, H 

Irradiancy, 1 or H 

— Radiant 


or R 

incidance 

Radiant 

Radiant intensity, J 

— Radiant 

intensity, J 


intensity 

Radiance, N 

Steradiancy or 

Steradiancy Radiant 


radiance, B slr 

sterance 


Radiant flux density 



(not source), W 


*Note Added in Proof: The Nomenclature Committee of the 

Optical Society of America has recently recommended 

that the flux density radiated from a source be called radiant exitance; the ratio of such flux from a sample to that 
of a blackbody be called emittance; and that of an opaque sample with perfect surfaces be called emissivity. 


RADIOMETRIC QUANTITIES, SYMBOLS, AND UNITS 7 

2.1.1. Radiometric Quantities as Field Concepts. The extension of the defini¬ 
tions of radiant emittance, radiant intensity, radiance, and irradiance to describe the 
properties of a radiant field, as well as the properties of a source, is of great utility. 
In particular, geometrically (neglecting attenuation by absorption, reflection, or scat¬ 
tering) the radiance at any point along a ray, in the direction of the ray, is invariant 
within an isotropic medium. In general, N/n 2 (where n is the index of refraction) is 
similarly invariant across a smooth boundary between two different media [5]. How¬ 
ever, the determination of the radiometric properties of a source in an attenuating 
medium, from measurements made at a distance, always involves some assumptions 
about the nature of the attenuation, emission, and scattering of the intervening medium. 
Thus, source characteristics calculated from field quantities may be in error by an 
unknown amount. If no attempt is made in the calculation of the source characteristics 
to include the effects of the intervening medium, then the calculated quantities should 
have the adjectival modifier apparent, or they should indicate that the radiant field 
at the point of measurement is described. 

2.1.2. Other Radiometric Quantities. Radiant absorptance, a, radiant reflectance, 
p, radiant transmittance, r, and radiant emittance, e, can be defined as: 


R 


a = 


absorbed 


P = 


R incident 
R reflected 


R 

R 


incident 


T = 


transmitted 
Rincident 

Remitted 


€ — 


R 


> 


blackbody ^ 


( 2 - 1 ) 


where R is the appropriate radiant quantity J, W, H, or N. 

Table 2-6 provides a list of definitions for the "other” radiometric properties. The 
symbolism is still in a transitory stage. The symbols k, T, H, N, etc., shown as sub¬ 
scripts (e t h and a s are also found elsewhere) do not indicate differentiation, e.g., ex ^ 
de/dk, and for this reason WGIRB has recommended the use of e(X) rather than ex 
for spectral emissivity, etc. 

Radiant absorptance should not be confused with absorption coefficient, which is 
often represented by the symbol a; however, the symbol a for absorption coefficient 
is preferred. 

The processes of absorption, reflection (including scattering), and transmission 
account for all incident radiation in any particular situation, and the absorptance, 
reflectance, and transmittance must add up to one: 

a + p + r = 1 (2-2) 

If a material is so opaque that it transmits no radiation, r = 0 and 

a + p = l (2-3) 

No specification has been made as to whether specular or diffuse quantities are in¬ 
dicated, and therefore the relations hold if the same specification is made for each of 
the quantities. 








8 


RADIATION THEORY 


e 


a 


P 

€, a, p 


T 


T 


Table 2-6. "Other” Radiometric Quantities* 


Emissance 


Absorptance 


Reflectance 


Emissivity, 

absorptivity, 

reflectivity 


Transmittance 


Transmissivity 


The ratio of the rate of radiant energy 
emission from a body, as a consequence 
of its temperature only, to the correspond¬ 
ing rate of emission from a blackbody at 
the same temperature. 

The ratio of the radiant energy absorbed 
by a body to that incident upon it. 

The ratio of the radiant energy reflected 
by a body to that incident upon it. 

Special cases of emissance, absorptance 
and reflectance; each is a fundamental 
property of a material that has an opti¬ 
cally smooth surface and is sufficiently 
thick to be opaque. 

The ratio of the radiant energy transmitted 
through a body to that incident upon it. 

Transmittance for a unit thickness sample. 


e, a, p, and r require additional qualifications for precise definition. The 
terms total and spectral, and the terms hemispherical, normal, and directional, 
are used and are indicated by subscripts, as illustrated here for emissance, e. 
In each case the emissance is the ratio of radiation from a surface, as a con¬ 
sequence of its temperature, to that from a blackbody at the same temperature. 
The subscript indicates the way(s) in which this radiation is limited as to 
wavelength and/or direction. 


€k 

Spectral 

emissance 

Ratio of spectral radiancy (or monochromatic 
radiancy at a given wavelength) from a 
body to that of a blackbody. 

€t 

Total emissance 

Ratio of total radiancy from a body to that 
of a blackbody. 


Hemispherical 

emissance 

Ratio of radiancy from a body to that of a 
blackbody. 

€0 

Directional 

emissance 

Ratio of steradiancy from a body to that of 
a blackbody. 

€n 

Normal 

emissance 

The special case of directional emissance 
when the emissance is in a direction normal 
to the surface. 

Therefore, precise definitions of emissance will take the following nomenclature: 

£th 

Total hemispherical emissance 

€t,\ 

Total normal emissance 


€\h 

Spectral hemispherical emissance 


Spectral normal emissance 


*These definitions follow the recent recommendation of NBS personnel wno propose the suffix -ance for specimen 
properties and -ivity for intrinsic material properties. Emissance follows the suggestion of Judd to avoid confusion 
with a power flux. The more recent OSA recommendation would change emissance to emittance. 


RADIATION LAWS 9 

2.2. Kirchhoff’s Law: Emissivity and Blackbodies 

Kirchhoff’s radiation law is 


- = W bb = W° 

a 


Kirchhoff’s law can also be stated as: 


(2-4) 


e = a 


(2-5) 


Thereby e is W/W f)b . 

For the limiting conditions of an opaque material (r = 0) to which Kirchhoff’s law 
applies: 


€ = 1 - p ( 2 - 6 ) 

A blackbody is a perfect absorber and therefore can be characterized by: 

a = € = 1, p = 0, t=0 (2-7) 

2.3. Radiation Laws 

Planck’s law is the basis for almost all radiometric considerations; other expressions 
are derivable from it. 

2.3.1. Planck’s Law. The blackbody spectral energy density for unpolarized radia¬ 
tion is given as 


Wy = 2t rc 2 h\-He hc ^ kT - l )- 1 

CC i 

W x = “■ \-*(e c * IXT — l ) -1 ( 2 - 8 ) 

This pair of equations also defines c i and c 2 as 

Ci = 87 rch 

c 2 = chlk (2-9) 

The literature sometimes contains alternative definitions for Ci that take geometrical 
factors into account. Usually Ci = 87 rch for the energy density expression, = 2nc 2 h 
for radiant emittance in a hemisphere, and = 2 c 2 h for radiance. The first (ci = 8nch) 
will be always used here. 

W\ can be related to u x as follows: 


u K = 4 Wylc 


( 2 - 10 a) 


where W K is the radiant emittance in a uniform enclosure; 


Uy = 2 Wy/C 

where Wy is the radiant emittance into a Lambertian hemisphere. 
Wavelength and frequency are related as follows: 


u v 



-X 2 

c 


( 2 - 10 b) 


—c 

uy =-7 u K 


( 2 - 11 ) 



10 


RADIATION THEORY 


Another useful form for Wx is 


Also, 


W x d\ = W v dv = 


2-n{kTY / x 3 dx \ 
c 2 h 3 \e x — 1/ 


x 


hv 

kT 


ttN\ = W\ 
4nNx = cu\ 


( 2 - 12 ) 

(2-13) 

(2-14) 

(2-15) 


2.3.2. Quantum Rates in Blackbody Radiation. The energy of a quantum is 

__ he (20 x 10 -20 joule /x) (1.2398 ev /*) 

C/= '“' = T = -x-- =-x ~ <2 - 16) 

The monochromatic flux n is 

/i„ = W v lhv = 2nc~ 2 v 2 (e hvlkT — 1) _1 photon cm -2 cps -1 (2-17) 

rix — WxX/hc = 27 Tc\~ 4 (e hclKkT — 1) _1 photon cm -3 (2-18) 

The mean square fluctuation rate is 

When e hvlkT is large enough, the classical result is obtained: 

(An) 2 = n (2-20) 

2.3.3. Stefan-Boltzmann Law. When the Planck equation is integrated over all 
wavelengths the Stefan-Boltzmann expression is obtained: 

W = crT 4 (2-21) 

where 

W is total radiant emittance, w cm -2 


a is the Stefan-Boltzmann constant, 5.67 x 10~ 12 w cm -2 (°K)~ 4 


T is absolute temperature 

2.3.4. Rayleigh-Jeans and Wien Laws. These two expressions antedated Planck’s 
formulation. They are incorrect but sometimes give useful approximations. 

The Rayleigh-Jeans equation is 

ux = c,\- 5 (A77c 2 ) = 87rkT\~ 4 (2-22) 


The Wien expression is 

ux = CiX ~ 5 e - hclXkT = 877 -cA - 5 e ~ hc,XkT (2-23) 

2.3.5. The Wien Displacement Law. This simple expression tells where the peak 
of the radiation curve falls at any given temperature: 


XrnaxT = 2897.9/* (°K) 


(2-24) 


2.3.6. Maximum Difference Expression. An expression for the wavelength at 
which the maximum monochromatic radiation difference for a given temperature 
difference occurs is found by writing 


d 3 W 
d 2 X dT 


= 0 


(2-25) 









BLACKBODY SLIDE RULES 


11 


The resultant condition is 

Xmax diff.T = 2404/li (°K) (2-26) 

2.4. Blackbody Slide Rules 

Rules have been devised for rapid, fairly accurate calculations of radiometric quan¬ 
tities. 

2.4.1. The General Electric Rule [6]. This rule, designated GEN-15C, is available 
from the General Electric Company, 1 River Road, Schenectady, New York; it costs 
about one dollar. Calculations which can be made on the rule are as follows (see 
Fig. 2-1): 

1. Conversions of temperatures among Celsius, Kelvin, Fahrenheit, and Rankine 
by setting the temperature on one scale and reading it on another — scales ABKL. 

2. Multiplication by the use of standard C and D log scales. 

3. Total blackbody radiant emittance by setting the temperature of the blackbody 
source on a temperature scale and reading on the E scale (w cm -2 ). An emissivity 
scale associated with the E scale permits direct calculation for graybodies; read 
the value on the E scale under the appropriate emissivity. 

4. Incremental blackbody radiant emittance Wax at maximum. The power density 
for a 1-/Li bandpass can be read directly from the Wx mnir or F scale. 

5. The ratio of W\ at any wavelength X to that at X mfl x, WJW), m(lx . The temper¬ 
ature is set on a temperature scale; then Wx/W\ max is read from the G scale 
opposite the desired X on the H scale. Thus one can find W\, Ha<r for a given X 
on the W\ max scale and then calculate the value of W\ at any wavelength on the 
WJW Xmax scale. 

6. The blackbody radiation in any spectral interval. Set the temperature scale 
at the appropriate temperature. Then on the W 0 -a/Wo-« or J scale read the 
percentage radiation that lies below a particular wave length Xi (on the I scale). 
Do the same for X 2 , and subtract. 

7. Conversion of range in nautical miles to range in centimeters with the aid of 
a straight edge, and vacuum calculation of irradiance. These can be made with 
the QRST scales. 

8. Conversion from w in. -2 to Btu ft -2 hr -1 . 

9. Number of photons sec -1 cm -2 from a blackbody at index temperature. 

Useful constants and other combinations of these calculations are also available. 

2.4.2. The Block Rule. This rule, available from Block Associates Inc., 385 Putnam 
Avenue, Cambridge 39, Massachusetts, at cost of about two dollars, consists of two 
curves of blackbody spectral radiance (Fig. 2-2); one is plotted as a function of wave¬ 
length and the other as a function of wavenumber. By shifting the curves along the 
lines marked, blackbody curves for any temperature can be obtained. The curves 
are made to cross the index line at the appropriate temperature. Then the values 
for the spectral radiance (unfortunately designated as 7x and /„) are read from the 
right-hand scales. On the reverse side a curve of Ay and AX vs X and v is given. Thus, 
if one knows a spectral resolution is A v at a given v he can find the resolution Ay at 
that y and the corresponding X. Of course, calculations from AX to Ay can also be 


12 


RADIATION THEORY 



Fig. 2-1. The GE slide rule. [General Electric Co., Light Military Electronics Dept., Utica, N.Y.] 












































BLACKBODY SLIDE RULES 


13 



i/i 5 

! i. 

2 £ 


< 


2 < 

2 >- - ^ - 

<§S§o 

i o I 5 i 

3 i I s I 

^ D ^ u. 

^ £ O 


Q 

E £ 

u r 


o 

Q 

Z 

o 

oc 

o 

5 


2 i I 


=/ II E 

,, O u 


“ o o 2 


! 

a 

i 


• O 

i z 
s < 

o Q 

V <3 o 

I •*- 


& 

5ft 

2S 


Q* ▼ Px co ▼ 


OD ~ ® 


ll 

U> «c 


£ => 
« —» 
* “■ 


5x1 
- 5 $ 


> 2 

>2 


D 3 ± ^ 

t- K U < 
< < V < 
a* oc 

£ £ ? 2 

5 S x 
¥->—¥- 
o o 
£ £ z z 

DD^ 

8 6 > > 
2 ^ < < 
< < * * 


w * 


• • 


OD Q 
a* * 

x ~ 2 | 

t- ■> N 


a £ 

% z 

E Z 


o z 
® < 
* £> 
^ 2 
< o 

v-» 


- z 

.. 3 


K 

C* 04 


5 


- 2 X 
O ? O 

5 I = 

< N u. 

“50 

>• Q ^ 


Q 2 ~ 

O O >= 
» un 


> z g 

s<o 
5 


< y «✓> u. 

►x 3? i*J .t4 


z 

II < X 

t s o 

N- 

tiU!- 
z z 
o;< 

U Z Q 

z < 

y"* < a£ 
U ^ -J 

< 3 o 


cu -- 

N-=i 


° -« £ 


O Ph 


D 

►- 

2 


W g J t 



0 - 
0 _ 

~s 

-cx 

§3 

-0 

“8 

§_= 

00 


2- 

—3 

O _ 
rx 

—00 

- : 

==f 

7 IN 

■y> 

rx_ 


Z 

i 


> 

s 


—<* p=" 


-x &- 


o = 


Sz= 


O _ 


'Vi 5 

iu * 


Sr 1 


> 

< 

* 


▲ ▲ 

* i 

l 't 

i 

zi 


A A 

«* >» Z £ 

s §gi 

> g-< ° 

Hi 


7 

9 

—- • 
n 

fr- I 

Z E 


* 

< 


17 

01 


E c 

u i 

« E 


n • 

I TO 


■* £ 

V? 

10 * 


= 

07 « 5 


< 

o 

tu 

IA 

k 

v 

z 
< 
—1 
o. 


o 

I 


u , 

< •h 

|i 


< 


< 

5 

M 

h- 

—1 

o 

CO 


V) 

Z 

o 


o 

m 


0» • -5 - X 

S * 5 » s 


: r« ^ 

• ±* * 
: ^ -e 

3 


^ 8 
* X 


lA 

c« 

« 


T 

n 

o> 

0 

0 

0 

0 

0 

0 

0 

X 

K 

M 

X 

X 

X 

X 

■o 

O 






CO 

00 


K 

K 

IV 

Ch 

O 

O 

m 

m 

O- 

•0 


O 

« 

n 

0 

K 

<x 

m 

m 

m 

•- 

n 

•- 

m 

* 


z 


* < 5 


in 

Z 

z 

>- 0 p 

Z 

Ui 

< 

u. 

Ill 

h- 

i/> 

< 

u. 

UJ 

fr¬ 

iz* 

5« sj 2 
Z S 3 0 

“ 3 8 a 

H- i^i K 


ui *- « k 
O < 5 2 

- i 


Fig. 2-1 (Continued). The GE slide rule. [General Electric Co., Light Military Electronics Dept., Utica, N.Y.] 



























































14 


RADIATION THEORY 



Fig. 2-2. The Block slide rule. [Block Associates, Inc., Cambridge, Mass.] 











BLACKBODY SLIDE RULES 


15 



Fig. 2-2 (Continued). The Block slide rule. [Block Associates, Inc., Cambridge, Mass.] 


















































































































































































































































































16 


RADIATION THEORY 



Fig. 2-3. The ARL slide rule. [Admiralty Research Laboratories, Teddington, Middlesex, England.] 




























































































































BLACKBODY CURVES 17 

made. Although the curves are correct, there are slight errors in the text: the table 
which is an insert to the spectral resolution calculator should have 


—kkv XAi> 

AX =-=- 

V 2 V 

(2-27) 

—&AX i^AX 

A v =-=- 

X 2 X 

(2-28) 


A useful way to remember this relation is 


AX _ —A^ 
X v 


(2-29) 


2.4.3. The Admiralty Research Laboratory Rule [7]. This more precise, more 
expensive (about $150) rule provides essentially the same calculations as the General 
Electric rule. It can be obtained from: ( 1 ) A. G. Thornton Company, Ltd., P.O. Box 3 , 
Wythenshawe, Manchester, England; (2) Jarrell Ash Company, Boston, Massachusetts; 
or (3) International Scientific and Precision Instrument Company, Inc., 910 Seventeenth 
Street N.W., Washington 6 , D.C. The symbol H, rather than W, is used for flux density 
(possibly indicating irradiance). The following calculations can be made (Fig. 2-3): 

1. The total irradiance (w cm -2 ) for a given temperature T can be read from scale 
a under a hairline set at temperature T on scale c (°C) or d (°K). 

2. The spectral irradiance for a 1 -cm wavelength spectral bandwidth at the maxi¬ 
mum of the curve H\ max (w cm -2 ) can be read from scale b under the hairline 
for the same setting of T on scales c or d. 

3. Similar quantities in terms of the number of photons Q can be obtained on scale 
f for Q (photons sec -1 cm -2 ) and on scale g for Q\ max (photons sec -1 cm -2 cm' 1 ). 

4. The ratio of H\ to H Kmnx can be found by placing the arrow marked TEMPERA¬ 
TURE at the appropriate temperature and using the hairline to find H x IH\ mnj . 
on scale a for X on scale i. Similarly, H\-JH can be read on scale i, Ho-JH on 
scale m, Q K /Q Kmax on scale n, Q\- x /Q on scale r, or Qo-JQ on scale s. 

5. Scale e is the wavelength scale on which one can also read wavelengths at which 
H\, Q\, H v , and Q v have their maximum values. 


The temperature scale runs from 100°K to 1 0,000° K, but if higher or lower temper¬ 
atures are desired, the rule can be extended by the use of the multiplication tables k 
and j. The instructions given here and more are provided on the reverse side of the 
rule, which also gives wavelength, wavenumber, and energy scales (but no hairline for 
conversion!). 

2.5. Blackbody Curves [8] 

Figures 2-4, 2-5, and 2-6 provide information about the spectral distribution of black¬ 
body radiation. The first of these is a linear curve with wavelength as the abscissa 
and W\ as the ordinate. The second is semilogarithmic but for a different range of 
temperatures. The third is a curve plotted on a logarithmic wavelength scale and 
linear emittance scale in terms of the variable \T. 

Another useful curve is a log-log plot of the Planck equation. The shape of the curve 
is identical for all T and need only be shifted along the line representing the Wien 
displacement law. Such curves are shown in Figs. 2-7 and 2-8. The straight lines 
are the "sliding lines.” Every blackbody curve for any temperature can be obtained 







18 


RADIATION THEORY 


l 

=1 

eg 

i 

£ 

CJ 

j* 

U 

u 

z 

< 

E- 

H 

»-* 

S 

w 

H 

z 

< 

Q 

< 

a 



WAVELENGTH (m) 

Fig. 2-4. Blackbody curves, 1000°K to 2000°K. 


a. 

eg 

i 

£ 

o 

£ 

W 

u 

z 

< 

H 

H 

s 

w 

H 

z 

< 

3 

2 



WAVELENGTH (m) 


Fig. 2-5. Blackbody curves, 100°K to 1000°K. 









BLACKBODY CURVES 


19 


FRACTION OF RADIANT EMITTANCE BELOW POINT INDICATED 

W /W 
0-X/ O-oo 





6 

o 




W 

o 

z 

< 

H 

H 

HH 

s 

w 

E-i 

Z 

S 

Q 

K 



10 6 10 5 10 4 10 3 10 2 

WAVE NUMBER (cm' 1 ) 


Fig. 2-7. A. and v. 


by moving a curve of the same shape along this line. Thus a "do-it-yourself” slide rule 
can be constructed by putting an overlay on this figure, tracing the curve and the line, 
and placing an index marker at 6000°K, the temperature of the top curve. Then by 
keeping the lines overlapped and setting the index marker at the desired temperature, 
the template becomes the blackbody curve for that temperature. Figure 2-8 is the 
same sort of curve but for a different temperature region. These are similar to the 
Block slide rule. 





20 


RADIATION THEORY 



WAVE NUMBER (cm *) 
Fig. 2-8. Wx vs A. and v. 



Fig. 2-9. Deviations of Wien and Rayleigh-Jeans 
expressions from Planck’s law. 


Often blackbody calculations can be made on the basis of either the Rayleigh-Jeans 


or the Wien expression. 

Thus, when hc/kkT is large, 



N x = 2 c 2 h\- 5 e~ hc ' XkT 

(2-30) 

When hc/\kT is small, 

Nx = 2 ck\~ 4 T 

(2-31) 

Figure 2-9 provides information about errors inherent in using these 

expressions. 







BLACKBODY TABLES 


21 


2.6. Blackbody Tables 

For very careful work it is necessary to use tables of blackbody functions with their 
additional precision and attendant difficulties. Since each set of tables is a thick 
book in itself, only references to these works are given here. 

S. A. Golden, Spectral and Integrated Blackbody Radiation Functions, Research 
Report 60-23, Rocketdyne Division, North American Aviation, Inc., Canoga Park, 
Calif. (1960) [9]. 

Table I provides WJW\ max and W 0 -k/W and their first derivative as functions 
of c 2 /XT. The intervals are as follows: 


c 2 /XT 


A c 2 /XT 


0-2 

2-5 

5-10 

10-25 

25-50 


0.01 

0.02 

0.05 

0.10 

0.20 


Table II provides W\ max and W as functions of T from 0°K to 10,000°K in 10°K 
intervals. Values of the radiation constants are 


Ci = 2nhc 2 = 3.7413 x 10 -5 erg cm 2 sec - 
c 2 = hc/k = 1.4388 cm °K 


M. Pivovonsky and M. Nagel, Tables of Blackbody Radiation Functions, Macmillan; 
New York (1961) [10]. 

Table I is a tabulation of (1) N\ vs X and T; (2) the ratio N\(\Ti)/N (0.560 g, 7\); 
(3) No-JN. These are tabulated to five significant figures for X = 0.2 g to 0.590 g 
in 0.005-g intervals and between 0.590 g and 1.2 g in 0.01-/U intervals from 800 
to 40,000° K in intervals varied to meet the needs of the range. 

Table II continues Table I for X = 1.1 g to 1100 g at temperatures from 20 to 
13,000°K with four-figure accuracy. 

Table III includes: (1) Nx/N\ max ; (2) a restatement of the wavelength ratios of 
Table I and II; and (3) a function for computing derivatives of the Planck function. 
These are plotted for XT from 0.01 to 0.99 g °K. Procedures for evaluation at 
higher values of XT are given. 

Table IV has: N, N\ max , X ma x for temperatures from 1000° to 2500°K at 2°K inter¬ 
vals, from 2500 to 5500°K in 5°K intervals and from 5500° to 10,000°K in 10°K 
intervals. 

Table V repeats Table IV but for wavenumbers, and Table VI repeats Table IV 
for reciprocal temperatures. 

Table VII gives luminance from 800° to 1796°K in 4°K intervals and a table of 
luminance and chromaticity coordinates. 

Table VIII is a temperature correction table —for revised physical constants. 

M. Czerny and A. Walther, Tables of the Fractional Function for the Planck Radi¬ 
ation Law, Springer-Verlag, Berlin (1961) [11]. 

Deals with W 0 -\/W plotted vs XT/c 2 . The tables are independent of c 2 . It also 
includes the first and second derivatives of this function and 



and its first derivative. 





22 


RADIATION THEORY 


Other tables in print are as follows: 

Parry Moon, J. Opt. Soc. Am. 38, 291 (1948) [12]. 

A. N. Lowen and G. Blanch, J. Opt. Soc. Am. 30, 70 (1940) [13]. 

E. Jahnke and F. Emde, Tables of Functions, Dover, New York (1945) [14]. 

A. G. DeBell, Rocketdyne Research Report 59-32, Rocketdyne Division of North 
American Aviation, Inc., Canoga Park, Calif. (1959) [15]. 

Still others exist for special uses: C. C. Ferriso of Convair prepared a set for flame 
calculations (in terms of wavenumber) and S. Twomey of NASA prepared a set (180° 
to 315°K and 20 cm -1 to 3400 cm 1 ), mostly useful for meteorological work. 

2.7. Radiation Geometry 


The differential solid angle is (see Fig. 2-10): 


dCl 


r 2 sin d dd d(f 
r 2 


(2-32) 


The solid angle of a sphere is Att sr. 

The solid angle of a hemisphere is 2 77 - sr. 

For small angles the solid angle is the product of the plane angles of two sides of an 
area, and is equal to the area divided by the distance squared. 



Fig. 2-10. Radiation geometry. 


2.7.1. Lambertian Sources. When radiance is defined, as in Table 2-1, as 


N = 


d 2 P 

COS ddA dfl 


w cnr 2 sr -1 


(2-33) 


a Lambertian surface has a constant radiance, which is the same in all directions. 

In general, when the radiance of a surface at a point is expressed as a function of 
direction, N = N(d, <p), the radiant emittance at that point is given by 


W 


N cos d dCl 


= aP = f 

~ dA~ J 

= J J N{d , <p) cos d sin d dd dip w cm -2 


(2-34) 


where the integration covers the entire solid angle containing the radiation beam of 
interest. If this is a perpendicular inverted circular cone of half-angle a. 



N(6 , (p ) cos 6 sin 6 dd dip 


w cm 2 


(2-35) 








DIRECTIONAL REFLECTANCE AND EMISSIVITY 


23 


If the surface is Lambertian (TV = constant), 

W = 2ttN I cos 0 sin 0 d6 = irN sin 2 a w cm -2 (2-36) 

Jo 

The radiant emittance into a hemisphere (a = n/2) from a Lambertian surface is then 

W = ttN wcm~ 2 (2-37) 

It is not 

W = 2 77 - TV w cm -2 (2-38) 

2.8. Distributed Radiators 

If a volume rather than a surface is the source of radiation, then when pressure and 
temperature along the path are constant, 

Wx = WV(1 - e~ axx ) (2-39) 

where W\ = spectral radiant emittance of the radiator 

Wx° = spectral radiant emittance of a blackbody 

a x = spectral absorption coefficient 

x = radiating path of the material 

The emissivity of such a partially transparent body is 

ex = 1 - e- a * (2-40) 

2.9. Selective Radiators 

The radiation from selective radiators can be calculated only by complicated processes. 
Below are listed some of the useful relations. Penner [3] is a good source for the theory. 

2.10. Directional Reflectance and Emissivity 

The basic interrelationships among the six quantities, normal and diffuse reflectance 
and transmittance, and hemispherical and directional emissivity, are concisely stated 
in an appendix to a paper by Richmond beginning on page 151 of [161. These relation¬ 
ships are not easily visualized, and some readers may find the following alternative 
approach and terminology helpful. Only opaque bodies of zero transmittance are 
treated. 

The radiance TV*, a function of both position and direction, is incident on the surface 
of an opaque body where some of the radiation is absorbed and the rest is reflected 
(as used here, reflected includes diffuse reflectance or scattering) to form a second radia¬ 
tion field, where the radiance N r of the reflected radiation is also a function of position 
and direction. N r is directly proportional to TV, in the sense that, if the value of TV* is 
multiplied by a constant that is independent of position and direction, the resulting 
values of TV r will all be multiplied by the same constant factor. However, it will be 
seen below that the interdependence of the spatial and directional distributions of 
TV r and TV* is more complex. 

The radiant power incident on a particular element 8A of the reflecting surface 
shown in Fig. 2-11, through an elementary beam of solid angle 80,, from a direction 
(0*, (p { ), is given by 

8P,(0 f , pi) — Ni(0i, pi) cos Qi 8A80< 

= TV,(0,, <p,)80',8A 


where 80', = cos 0, 80 

= sin 0, cos 0, ddi dpi. 


w 


(2-41) 


24 


RADIATION THEORY 


z 



Fig. 2-11. Geometry of incident and 
reflected elementary beams. The Z 
axis is chosen along the normal to 
the surface element at O. 


The quantity SO'j is the projected solid angle [5,17] of the elementary beam. Cor¬ 
respondingly, the irradiance at 8 A is 

SHiidi, (fi) = Ni(6i, (pi)8Cl'i wcm -2 (2-42) 


Then the radiant intensity of the surface element 8A, due to reflection (scattering) of 
radiation from this incident elementary beam, in the direction (0 r ,<pr) is 

8Jr(6r, (fr) = p'(0,, (pi , Or , <pr) COS 0 r 8Pi(0;, (pi) w sr- 1 (2-43) 

and the reflected (scattered) radiance 


where 


8N r (0r, (fr) = p'(0i, (pi, 0 r , (p r )8Hi(0i , <pi) 


w cm -2 sr -1 


= 8N r (0r, <f>r) _ 8N r (0r , <f>r) 

~ 8Hi{0j , 4>i) ~ Ni(0i, 4>i)8Cl'i 


sr _1 


(2-44) 

(2-45) 


p' is the partial reflectance, or reflection-distribution function [18] of the surface ele¬ 
ment 8A for radiation incident from the direction (0,, </>,) and reflected (scattered) 
in the direction ( 6 r , <M- Furthermore, by a reciprocity theorem of wide generality 
[19, 20] first enunciated by Helmholtz:* 

p'(0i, (f) U 02, </> 2 ) = p'(02, <t> 2 , 0i, </>i) sr- 1 (2-46) 


Thus p'(0i,4>i, 0 2 , </> 2 ) is the partial reflectance between the two directions (0i,</>j) 
and (0 2 , 0 2 ), where either direction may be that of the incident elementary beam and 
the other that of the reflected (scattered) elementary beam. 

Hence the radiance at a point of the reflecting surface (taken as the origin for spheri¬ 
cal coordinates) in the direction (0 r , <p r ) and due to reflection (scattering) of all beams 
of incident radiation is 


*A search for a proof (in English) of this important theorem also turned up a number of authors 
who referred to or made use of the theorem in various ways without giving a proof [21-24], includ¬ 
ing von Helmholtz himself [25], although Planck [26] states, without specific citation, that von 
Helmholtz "proved” the theorem. DeHoop [20] not only gives a proof (essentially the same as that 
of Kerr [19]) but also includes an explicit statement of the requisite conditions. 









DIRECTIONAL REFLECTANCE AND EMISSIVITY 


25 


J ’2it 

p'Ni sin di cos Si ddi d(f>i 

o Jo 


dfl'i w cm -2 sr -1 


- L p ' Ni 

The following notation is used to designate integration over a hemisphere: 


(2-47) 


J p f 2 tt Ctrl 2 

f[6,(p)dCl= f{d , <p) sin S dd d<p 

h Jo Jo 


and 


f f{6,(p)dCl'= f f f{0, <p) sin 0 cos 8 d0 d<p 

Jh Jo Jo 


Relation (2-47) is for a particular point, or for the surface element 8A at that point. 
For a more general expression, one must also establish the reflected radiance from other 
points. When p' and iV, are expressed as functions of spatial location (as well as di¬ 
rection) for all points on the reflecting surface, Eq. (2-47) gives the reflected radiance 
N r as a function of position for these same points on the reflecting surface, as well as 
for direction (0 r , <p r ) at each such point. It is important here to recognize that Eq. 
(2-47) is written above in coordinates which, for convenience, are specially oriented 
with respect to the surface element 8A. Appropriate adjustments must be made when 
dealing with irregular surfaces where the direction of the normal changes in going 
from one surface element to another. 

Whether surface irregularities are treated as microscopic [in the sense that their 
effects are integrated or averaged in the distribution function or partial reflectance 
p'(0i, <Pi, 02, <p>)], or as macroscopic (in the sense that they may be analyzed into 
smaller surface elements 8A for treatment as above) can be arbitrary, depending on 
the degree of resolution desired, or can be dependent on circumstances limiting achiev¬ 
able resolution. For example, in examining the reflectance of a highly irregular sur¬ 
face containing deep cavities, such as a piece of volcanic scoria, or a coarse, blackened 
cellulose sponge in the laboratory, it may be possible to consider the reflectance of 
different portions of the walls of single cavities (which are then regarded as macroscopic 
irregularities). But when one studies the possible effects of similar surfaces which 
may exist on the moon, where such fine detail cannot possibly be resolved by the best 
telescopes on earth, these are necessarily treated as microscopic irregularities [27,28]. 
Still more complicated considerations are introduced when microscopic irregularities 
are small enough to have dimensions of about, or less than, the wavelength of the 
incident light or other electromagnetic radiation [29-32]. 

The total reflectance p of a surface element 8A is defined in general as 

p = 8P r /8Pi dimensionless (2-48) 

where 8P, is the total radiant power incident (from all directions) on 8A, and 8P r is 
the total resulting reflected radiant power (in all directions). As stated above, the 
value of p depends on the geometry and spectrum of the incident beam of radiation, 
which may be different in each particular case. Here, for the moment, the primary 
concern is the geometry. Hence spectral considerations will be eliminated for the 
remainder of this section by restricting the spectrum of the incident radiation, except 
where otherwise stated, to a region over which p does not change significantly with 
wavelength. It is then useful to consider some special cases of incident-beam geom¬ 
etry. 


26 


RADIATION THEORY 


If the incident radiation is well collimated, within a small element of solid angle 80, 
= sin Oiddi d(fi from the direction (0,-, y?,), the total radiant power incident on 8A is 

8P, = 8Hi(di,<pi)8A w (2-49) 

Then, from Eq. (2-45), 


But 


8N r (0 r , (Pr ) = p' 8Hi(6i, </>,) 

= p' 8Pi/8A w cm -2 sr -1 

8Pr = 8A J h 8Nr(dr , <?r) ftfVr 

= 8P, |^ p'(0t, Or, (pr ) rfO'r 
= 8PiPdi(0i, (pi) w 


(2-50) 


(2-51) 


where p d i(0,, <p<) is the (total) directional reflectance for a well-collimated incident beam: 

PdiiOi, <pi) = p'(di, (pt, d r , <p r ) dCl' r dimensionless (2-52) 

Jh 

For isotropic surfaces, there is no dependence on the azimuth y>, and Eq. (2-52) simpli¬ 
fies to the frequently recognized dependence on 0: pd,(0,, y?,) = p d ,(0i). If the well- 
collimated beam is incident perpendicularly on a plane surface, this becomes the com¬ 
monly reported normal reflectance p n — Pdi(O). If a point on the surface of an opaque 
solid is uniformly irradiated from all external directions, i.e., if Ni is a constant, the 
reflected radiance in the direction (0 r , <p r ), from Eq. (2-47) is given by 

N r (Or, (pr) = Ni f p' dft'f 

= NiPdASr, (pr) w cm -2 sr -1 (2-53) 

where 

Pdr(0r i <Pr) ~ I, p'(di , y>i, 6 r , <p r ) dCt'i dimensionless (2-54) 

But, from the reciprocity relation, Eq. (2-46), and Eqs. (2-52) and (2-54), 

Pdi(0i, (Pi) = Pdr(0i, (pi) = pd(0\, (pi) dimensionless (2-55) 

Thus the (total) directional reflectance pd(0i, y?i) for a well-collimated beam incident 
from the direction (0i, y>i) is also the ratio between the reflected radiance iV r (0j, y?i) 
in that same direction and the incident radiance Ni when the surface is uniformly 
irradiated from all directions (hemispherical irradiation). This relation [Eqs. (2-53) 
and (2-55)] is the basis for a reflectometry technique described by McNicholas [22]. 

More important, Eqs. (2-51), (2-53), and (2-55) are the basis for evaluating and 
equating the directional absorptance and directional emissivity of the surface ele¬ 
ment 8A in a simple relation which has the same form as the Kirchhoff’s law relation, 
Eq. (2-6). If, in Eq. (2-53), the uniform incident radiance Ni is equal to NbiT), the 
blackbody radiance (either total or spectral, i.e., in a small wavelength interval at a 
given wavelength) in an isothermal enclosure at T°K, and if, in fact, the reflecting 
surface forms the wall of such an enclosure so that it too is at this same temperature, 
then the radiance in the direction (0 1 , y?i) from the element of wall surface 8A is made 
up of an emitted radiance and a reflected radiance, as follows: 


Ne + Nr = €d(di, <pi)Nb(T) + Pdr(ei,<Pi)N b {T) = N b (T) 


w cm -2 sr -1 (2-56a) 


DIRECTIONAL REFLECTANCE AND EMISSIVITY 


27 

Similarly, the radiance from the direction (0i, <pi) incident on the element 8A is made 
up of an absorbed incident radiance and a reflected incident radiance (scattered in 
all directions): 

N ia + N ir = ad{0u (f>i)Nb(T) + pdi( 6 1 , <pi)N b (T) 

= N b (T) w cm~ 2 sr _1 (2-56b) 

Here, € d (0 1 , <p i) is the directional emissivity (at temperature T ) of the element 8A for 
radiation emitted in the direction ( di,<p \) and ad(0\,<pi) is the absorptance (at T ) 
for radiation incident from that direction. Consequently, from Eq. (2-55), 

€d(#l, </>l) = 1 — Pdr ( 0 1, <£i) 


= 1 — pdi(0 1 , </>i) = acdid i, </>i) dimensionless (2-57) 

Note that equilibrium maintenance with conservation of energy (Kirchhoff’s law) 
by itself would justify only each line of Eq. (2-57) independently, and the Helmholtz 
Reciprocity law (which is the basis for Eq. (2-46) and, in turn, Eq. (2-55)) must also 
be invoked in order to equate them to each other and so to relate emissivity for radi¬ 
ation emitted into a given direction to the absorptance for radiation incident from 
that same direction. 

In the more familiar form of Kirchhoff’s law, 


e = 1 — p = a dimensionless 


( 2 - 6 ) 


directional quantities are not considered. Instead, the total emissivity for radiation 
emitted in all directions (into a hemisphere) is related to the total reflectance (in all 
directions into a hemisphere) for uniform incident radiance (from all directions, i.e., 
from a hemisphere) and to the total absorptance for uniform incident radiance (from 
all directions, i.e., from a hemisphere). The total reflectance p in Eq. (2-6), for uni¬ 
form incident radiance (iV, = a constant independent of direction) is then 


8A N r 8a' 

p = 8P r l8Pi =-y- 

8A Ni8n' 



da' 




dimensionless 


(2-59) 


The quantities in Eq. (2-6) are those involved in heat-transfer computations where 
the interest is in the net flow of energy across a bounding surface, involving radiation 
received, emitted, or reflected in all directions. 

Equations (2-6) and (2-57) apply in all cases to spectral radiation (i.e., the radiation 
in a very small wavelength interval about a specified wavelength) and hence also to 
any spectral interval in which p or pd (and therefore also e or €</ and a or ad) do not 
change significantly with wavelength. When thermal equilibrium exists, i.e., when 


Ni = N b (T) = r Nxb(T,\)d\ 
Jo 


where N\ b (T, X) is the spectral radiance of a blackbody at 'T’K, they also apply to total 
radiation (all wavelengths), even though the spectral reflectance varies with wave¬ 
length. However, if the spectral reflectance is not a constant and the spectral distri¬ 
bution of the incident radiation is arbitrary (nonequilibrium condition), Eq. (2-6) 




28 


RADIATION THEORY 


and (2-57) do not necessarily hold for the total (all wavelengths) reflectance, absorp- 
tance, and emissivity. 

The significance of the partial reflectance p' and directional reflectance pd can be 
clarified by relating them to the more familiar ideas of diffuse and specular reflectance. 
First, a perfectly diffuse reflector is one for which p' = a constant, so that 

Pd = p' J dSl' = 7 Tp' dimensionless (2-60) 

Hence pd is also a constant, i.e., the same in all directions, so the total reflectance p = pd = 
np ' for any arbitrary configuration of incident radiation. Second, a perfectly specular 
reflector is characterized by 

N r (0, <p±tt) = pd(0, <p) w cm -2 sr 1 (2-61) 

By comparing this with the general relationship between incident and reflected radi¬ 
ances, it can be seen that Eq. (2-61) will result if the partial reflectance p' in Eq. (2-47) 
has the form 

p' = 2 p d (0i, <pi )8(sin 2 d r ~ sin 2 6i)8((p r — <fi±n ) sr -1 (2-62) 

where 8(sin 2 6 r — sin 2 0,) and 8((p r — <pi±n) are Dirac delta functions which satisfy the 
defining relations 

8(w) = 0 for u ^ 0 
I 8(u) du = 1 

and 


J f(u)8(u) du = f( 0 ) 

when the integration is carried out over the full range of the variable, 0 *£ 0 ^ tt/2 
and 0 «£ cp =s 2-7T, in each case. 

Only what might be termed the external radiometric relations have been considered 
in the foregoing treatment, and no attempt has been made to deal with the deeper 
theory relating reflectance, emissivity, and absorptance to the optical constants of the 
materials. A good summary of the most important aspects of that approach is given 
in [331. 

2.11. Summary of Equations and Constants 

W = aT 4 (2-21) 

W x = 2t TC 2 hk-s(c hc ' kT - l)" 1 (2-8) 

W v ~ 2v■c~ 2 hv 3 (e hvlkT — l) -1 [follows from (2-8) and (2-11)] 

N\ = W\ln (2-19) (only Lambertian surfaces, e.g., blackbody) 

U\ = 1.2398 ev g. (2-14) 

u\ = 4 WJc (2-10a) (within isothermal enclosure) 

u\ = 2 Wxlc (2-106) (into Lambertian hemisphere) 

n v = WJ hu = 27rc~ 2 v 2 (e hvlkT — l) -1 (2-15) 

n x = Wx/hc = 2TTc\~ 4 (e hclkkT - l)" 1 (2-16) 

Wx d\ = W v dv (2-12) 

(An) 2 = n[l + ( e hvlkT — 1) _1 ] = n 
\maxT = 2897.9 (^ °K) (2-24) 


(2-17) and (2-18) 



kmax diff. T— 2404 ( M °K) 
c — vk 


REFERENCES 


29 


(2-26) 


cr — v — k/2i t = 1 Ik (wavenumber) 
dv = —c dk/k 2 

dk = —c c/W^ 2 
d\/X = dv/v 


J = P n = dP/dfl 


W = P A 

h = p a 


= dP/dA 


n = p a{1 


8 2 P 

cos o dA an 


a 2 P 

aA an' 


where dCF = cos 0 <in 


/i = 6.6252 x 10 -34 w sec 2 
tt = 3.1416 


c = 2.99793 X 10 8 m sec -1 


2tt 


hU4 


a = 


5.6686 X 10~ 12 watt cm -2 (°K) 4 (Stefan-Boltzmann constant) 


k 

e 

hlk 


15 c 2 h 3 

1.38047 x 10 16 erg (°K) _1 (Boltzmann’s constant) 
2.71828 (base of Napierian logarithms) 

4.079 x 10- 11 sec °K 


Quantity 

Ci 

Ci 

Energy density 

SttcH 

4.99 joule g 4 mr 3 

Emittance 

2ttc 2 h 

3.7413 x 10 8 w p, 4 m 

Radiance 

2 c 2 h 

1.19 X 10 8 w p. 4 m 2 

= chlk = 1.4388 x 10 4 

gL °K 



References 

1 . M. Holter et al., Fundamentals of Infrared Technology, Chap. 1, Macmillan, New York (1963). 

2. Report of the Working Group on Infrared Backgrounds, Part II:"Concepts and Units for the 

Presentation of Infrared Background Information,” Report No. 2389-3-S, The University of 
Michigan, Institute of Science and Technology, Ann Arbor, Mich. (1956) AD 123 097. 

3. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, 
Reading, Mass. (1959). 

4. G. Worthing and D. Halliday, Heat, Wiley, New York, 435 (1948). 

5. F. E. Nicodemus, "Radiance,” Am. J. Phys., 31, 5, 368 (1963). 

6 . A. H. Canada, Gen. Elec. Rev., 51, 50-54 (1948). 

7. M. W. Makowski, Rev. Sci. Inst., 20, 876 (1945). 

8 . T. P. Merritt and F. F. Hall, Jr., Proc. IRE Sept. (1959). 

9. S. A. Golden, Spectral and Integrated Blackbody Radiation Functions, Research Report 60-23, 

Pocketdyne Division, North American Aviation, Inc., Canoga Park, Calif. (1960). 

10. M. Pivovonsky and M. Nagel, Tables of Blackbody Radiation Functions, Macmillan, New 
York (1961). 

11. M. Czerny and A. Walther, Tables of the Fractional Function for the Planck Radiation Law, 
Springer-Verlag, Berlin (1961). 

12. Parry Moon, J. Opt. Soc. Am., 38, 291 (1948). 

13. A. N. Lowen and G. Blanch, J. Opt. Soc. Am., 30, 70 (1940). 

14. E. Jahnke and F. Emde, Tables of Functions, Dover, New York (1945). 






30 RADIATION THEORY 

15. A. G. DeBell, Rocketdyne Research Report 59-32, Rocketdyne Division, North American 
Aviation, Inc., Canoga Park, Calif. (1959). 

16. Henry Blau and Heinz Fischer (eds.), Radiative Transfer from Solid Materials, Macmillan, 
New York (1962). 

17. R. Clark Jones, "Immersed Radiation Detectors,” Appl. Opt., 1, 607 (1962). 

18. D. K. Edwards, J. T. Gier, K. E. Nelson, and R. D. Roddick, "Integrating Sphere for Imperfectly 
Diffuse Samples,” J. Opt. Soc. Am., 51, 1279 (1961). 

19. D. E. Kerr, "Application of the Lorentz Reciprocity Theorem to Scattering” Appendix A (p. 693) 
of Propagation of Short Radio Waves, Vol. 13 of M.I.T. Radiation Laboratory Series (McGraw- 
Hill, New York, 1951), first edition. 

20. A. T. DeHoop "A Reciprocity Theorem for the Electromagnetic Field Scattered by an Obstacle,” 
Appl. Sci. Res., Sec. B, 8, 135 (1960). 

21. H. J. McNicholas, "Absolute Methods in Reflectometry,” Bu. Std. J. Res., I, 29 (RP-3), submitted 
to The Johns Hopkins University, PhD, dissertation (1928). 

22. J. C. DeVos, "Evaluation of the Quality of a Blackbody,” Physica, 20, 669 (1954). 

23. Max Born and Emil Wolf, Principles of Optics, 380, Pergamon, New York (1959). 

24. Committee on Colorimetry, Optical Society of America, "The Science of Color,” 178, Crowell, 
New York (1954). 

25. H. von Helmholtz, "Helmholtz’s Treatise on Physiological Optics,” James P. C. Southall (ed.) 
translated from Third German Edition, Optical Society of America, I, 231, (1924). 

26. M. Planck, Theory of Heat, translated by H. L. Brose, 5, 198, Macmillan, New York (1957). 

27. Bruce Hapke and Hugh Van Horn, "Photometric Studies of Complex Surfaces, with Appli¬ 
cations to the Moon,” J. Geophys. Res. 68, 15, 4545 (Aug. 1, 1963). 

28. Bruce W. Hapke, "A Theoretical Photometric Function for the Lunar Surface,” J. Geophys. 
Res. 68, 15, 4571 (Aug. 1, 1963). 

29. V. Twersky, IRE Trans. Antennas and Propagation, AP-5, 81 (1957). 

30. V. Twersky, J. Res. NBS, 64D, 715 (1960). 

31. V. Twersky, "Multiple Scattering of Waves and Optical Phenomena” J. Opt. Soc. Am., 52, 
145 (1962). 

32. H. E. Bennett and J. O. Porteus, "Relation Between Surface Roughness and Specular Reflec¬ 
tance at Normal Incidence,” J. Opt. Soc. Am., 51, 123 (Feb. 1961). 

33. Henry H. Blau, Jr., John L. Miles, and Leland E. Ashman, Thermal Radiation Characteristics 
of Solid Materials — a Review, Arthur D. Little, Inc. Contract No. AF 19 (604)-2639, Scientific 
Report No. 1, AFCRC-TN-58-132 (1958) AD 146 883. 


Chapter 3 

ARTIFICIAL SOURCES* 


CONTENTS 


3.1. Theory.32 

3.1.1. The Method of Gouffe.32 

3.1.2. The Method of DeVos.33 

3.1.3. The Method of Sparrow.37 

3.2. NBS Standards.38 

3.2.1. Instructions for Using the Total Radiation Standards.38 

3.2.2. Instructions for Using the NBS Standards of Spectral Irradiance . . 40 

3.2.3. Instructions for Using Standards of Spectral Radiance.41 

3.2.4. Other NBS Sources. 44 

3.3. Laboratory and Field Sources.46 

3.3.1. A High-Temperature Source.46 

3.3.2. Low-Temperature Large-Area Sources.47 

3.3.3. Nernst Glower.48 

3.3.4. Globar.48 

3.3.5. Welsbach Mantle.48 

3.3.6. Carbon Arc.49 

3.3.7. Tungsten Filaments.49 

3.3.8. Mercury Arcs.50 

3.3.9. Zirconium Point Sources.51 

3.4. Commercial Cavity-Type Sources.51 


*Material prepared by the technical writing staff of McGraw-Hill, Inc. 


31 























3. Artificial Sources 


3.1. Theory 

The classic blackbody simulator is a spherical cavity made of an opaque material 
and pierced by a small aperture. If the enclosure is insulated from outside thermal 
influences, all parts of the internal cavity walls eventually reach the same temperature. 
When equilibrium is reached, all points inside the cavity have equal energy regardless 
of the nature of the walls, and the radiation passing out of the small aperture approxi¬ 
mates blackbody radiation; the smaller the hole the better the approximation. 

A practical cavity approaches a blackbody to the extent that the radiation entering 
through the aperture is absorbed within the cavity. Since all practical materials have 
absorptivities less than unity, the performance of a blackbody-simulating cavity is 
based on the number of reflections that radiation entering the cavity through the 
aperture can make before returning through the aperture to the outside. 

DeVos [1], Edwards [21, Gouffe [3], Williams [4], Sparrow [5], and Zissis [6] have 
written theoretical analyses of cavity radiation. Williams’ paper contains a criticism 
of the others. The most thorough treatment seems to be that of DeVos, although 
Williams’ criticisms of it are valid. Truenfels [7] has also written a pertinent article. 
Three techniques are given below (DeVos, Gouffe, and Sparrow). No direct detailed 
comparison of them is available. 

3.1.1. The Method of Gouffe. For the total emissivity of the cavity forming a black¬ 
body, Gouffe gives: 

e 0 = €'o(l + &) (3-1) 

where 

e 

6 ° = e[ 1 - (s/S)] + (s/S) (3_2) 

and k — (1 — e) [(s/S) — (s/S 0 )], and is always nearly zero; it can be either positive or 
negative 

e = emissivity of materials forming the blackbody surface 
s = area of aperture 
S = area of interior surface 

So = the surface of a sphere of the same depth as the cavity in the direction normal 
to the aperture 

Figure 3-1 is a graph for determining the emissivities of cavities with simple geomet¬ 
ric shapes. (Values not legible can be calculated without great difficulty.) In the lower 
section, the value of the ratio s/S as a function of the ratio L/R is read (L and R are 
defined in Fig. 3-1). Reading up from this value to the value of the intrinsic emissivity 
of the cavity material, the value of e' 0 is found. Multiplying e' 0 by the factor (1 + k) 
then yields the emissivity of the cavity. Values of s/S, etc., are given in Table 3-1. 

WTien the aperture diameter is smaller than the interior diameter of the cylindrical 
cavity or the base diameter of the cone forming a conical cavity, it is necessary to 
multiply the values of s/S determined from the graph by ( r/R ) 2 , which is the ratio of 
the squares of the aperture and cavity radii (Fig. 3-1). 


32 



THEORY 


33 



3.1.2. The Method of DeVos. DeVos considers a cavity of arbitrary shape, with 
opaque walls, in a nonattenuating medium, initially at a uniform steady temperature, 
with one small opening. He adds additional openings and temperature variations along 
the cavity walls, and indicates several practical approximations which are necessary 
for calculation of numerical values. See Fig. 3-2 for definitions of terms. 

The power emitted from dO is 

em Pw° = €«,°(A, T)Nx,b(K T) dw cos 0u,° d(l w ° (3-3) 

where e w °(\, T ) = the spectral emissivity of dw in the direction of dO (indicated by sub- 

and superscripts throughout) for temperature T and at wavelength A 

iVx,B(A, T) = the spectral radiance of a blackbody for temperature T and wave¬ 
length A, given by either modification of Eq. (3-2) or approximately 
by the Wien law, 

N\, b = (constant) e~ c * IXT 


(3-4) 






































































34 


ARTIFICIAL SOURCES 



Fig. 3-2. Definition of terms for the DeVos method. 


dw = area of the emitting infinitesimal element 

6 W ° = the angle of the direction from dw to dO with respect to the normal to dw 

dCl w ° = the solid angle subtended by dO, the hole, as seen from dw 

The power from dw through dO which is due to the reflection of the power received 
at dw from some arbitrary elemental wall area dn is 

refip w no = Nx,n w (k, T) dCl w n dw cos d w n r«, n 0 (A, T ) dQ w ° ( 3 - 5 ) 


where P w n0 = the power from dn to dO via dw 

N\, n w (\, T) = the spectral radiance from dn to dw for a temperature T and at 
wavelength \ 

dfl it" = the solid angle subtended by dn as seen from dw 

d w n = the angle to the normal to dw made by the direction from dw to dn 

r w n0 (\, T) = the partial reflectivity of dw for radiation from dn at 0 *. n reflected 
from dw toward dO at d w °, at wavelength A. and for temperature T 

Partial reflectivity can be defined as follows: 


(3-6) 


As shown in Fig. 3-2, the reflected rays are not necessarily in the same plane. How¬ 
ever, symmetry of the partial reflectivity about the plane containing the normal and 
the incident direction is assumed. 

N : n0 

r w n0 = —p— cos 6 W ° ( 3 . 7 ) 

it u' 

To obtain the power from dw due to the reflection from dw of the radiation from all 
parts of the cavity walls except dO, integrate Eq. (3-5) over the walls excluding dO: 

re f l P w ° = dw dSl w ° [ Nx,n w (\, T) cos 6 «,"r»°"(\, T) dn w " (3-8) 

Jail dn 


The reciprocity relation is: 


r ab cos d a = r* a cos 6 b 


(3-9) 








THEORY 


35 


By using the reciprocity relation in Eq. (3-9): 

refl Pw° = dw dn w 0 cos 8 w o I JV x .»»(X, Dr. # *(X, T) dCl w n 
To a first-order approximation the following relationship is true: 


Thus 


Therefore 


N x , n ”(\,T) = N K , B (k, T) 

r»°*(X, T) dO** = P tt ,° (X, T) - r»«(X, T) d(l w ° 

all dn 

t°taip u o _ rfic cos 0„,° dn«°(l — r M .°° rffl u ,°) 


(3-10) 


(3-11) 


(3-12) 


The hole can be considered to have an emissivity given by 

e 0 = 1 - r«,» d(l w ° (3-13) 

to a first approximation. If additional holes exist, or an opening like a slit which 
should be considered as several holes, then the reflected contributions of these elements 
must also be excluded. This leads to: 


€<>=1-2 r » oh (3-14) 

h 

for the emissivity of dO in the direction from dw when there exists several holes num¬ 
bered from h = 0 to some finite integer. 

For the second-order approximation DeVos considers the use of a value for N\, n w 
which is not N\, B , but which is calculated by considering the effects of the holes on 
this spectral radiance (from each element dn) and the effect of T n ¥= T w . The calcu¬ 
lation neglects variation, due to temperature differences, of the emissivities and re¬ 
flectivities of the wall elements. 

The second-order approximation for the quantity total P w ° is 


total P w ° = N\' B (\, T) dw cos 6 W ° dfl w ° 




r wh 
' n 


dCl n h r w 0n dCl w n 



k n € 


w r w 0n dPl 



(3-15) 


where the integration is over the entire surface excluding the holes. 

DeVos uses Wien’s equation to evaluate k n for cases where T — T n << T. Thus, 


and 


N\, B ~ e C * IXT 

dN\, B _ C 2 dT 
N x , b 


(3-16) 

(3-17) 


where the constant C 2 = hc/k for T in °K and X in cm. Thus 





36 


ARTIFICIAL SOURCES 


DeVos applied this theory to the V-wedge, cylindrical (closed at one end), spherical, 
and tubular shapes. He did not treat the cone, a combination of cones, or a cone- 
cylinder. He included temperature gradients only in the tubular calculation. 

For a cylinder of radius R and length L (see Fig. 3-1), the value of e 0 to first-order 
approximation is: 

(3-19) 


where a is the cone angle. 

For the second-order approximation (neglecting temperature gradients), one needs 
dCl w n and dfl u ,°. If dn is an annulus of the cylinder with a length dx, then 


and 


d£l w n = dCl w x 


27 rr 2 dx 

[( L-xY + r 2 ] 3 ' 2 


(3-20) 


da x ° 


nr 2 

x 2 4- r 2 


(3-21) 


if x/r = y, then 


€o — 1 


r a 

- 2- 2 ” 2 
a 2 Jo 


0 y r U)0 

') ' y 


(y 2 + 1) [(y - a) 2 + 1] 


3/2 


dy 


(3-22) 


DeVos evaluated this expression by numerical integration. His values, corrected 
by Edwards [2] for a small numerical error, are given in Table 3-1. For a similar 
calculation for a sphere, DeVos obtained the results in Table 3-2. 

Table 3-1. DeVos’ Emissivities, Cylindrical Blackbody [ 1 ] 

(Emissivity values for a cylindrical blackbody with second-order correc¬ 
tions for various values of a (= Dir = depth of cylinder/radius of cylinder) 
and surfaces of different smoothness. These are DeVos’ values corrected 
for a numerical error. DeVos’ figure numbers head the columns.) 


a 

Fig. 2 

Fig. 3 

Fig. 4 

Fig. 5 

6 

0.970 

0.954 

0.865 

0.668 

10 

0.990 

0.985 

0.953 

0.864 

15 

0.995 

0.994 

0.980 

0.947 

20 

0.997 

0.997 

0.989 

0.972 

30 

0.999 

0.999 

0.996 

0.988 


Table 3-2. DeVos’ Emissivities, Spherical Blackbody [1] 

(Emissivity values for a spherical blackbody with second-order correc¬ 
tions for various values of a {= D/r = diameter of sphere/radius of hole) 
and surfaces of different smoothness. These are DeVos’ values corrected 
for a numerical error. DeVos’ figure numbers head the the columns.) 


a 

Fig. 2 

Fig. 3 

Fig. 4 

Fig. 5 

10 

0.992 

0.989 

0.963 

0.894 

20 

0.998 

0.998 

0.991 

0.976 






THEORY 


37 


3.1.3. The Method of Sparrow [8]. "The starting point of the analysis is a 
radiant flux balance at a typical position x 0 on the cylindrical wall. Such a balance 
states that the radiant energy leaving a location is equal to the emission plus the re¬ 
flected incident energy. 


€a(*'o) = € + (1 — e) 


CLjd 

Jo 


€„(*') 1 


2(x' - x' 0 ) 2 -I- 3 ) 

2 [(x' — x' 0 ) 2 + 1] 3/2 J 


dx' 



(3-23) 


€a(r') = 


€+ 8 ( 1 - 6 ) 



«.(*') 



4 [l -*')* + 1 " ^ 



r /L v i 

2 3/2 

{ 

l 4 (a-* 1 ) +1 H 

- 4r' 2 


dx' 


(3-24) 


where e a = N/crT 4 , x' = xld, r' = r/i?. 

"It may be seen that e a (x') and e„(r') appear in both Eqs. (3-23) and (3-24), which 
therefore require simultaneous solution. Additionally, since the unknowns 
appear under integral signs, these equations are integral equations. Further 
inspection reveals that there are two parameters which may be independently 
prescribed: Lid, which describes the relative length of the hole; and e, which 
characterizes the radiation properties of the materials. 

"The mathematical problem as represented by these equations is complex, but 
this is the price which must be paid in order to achieve a complete formulation of 
the physical occurrences within the radiating enclosure... . By assuming c a (r') = 
constant, the system was reduced from two simultaneous integral equations to a 
single integral equation. Further, all angle factors were written approximately 
in terms of exponentials of the form e~\ x o~ x \, etc., rather than as the precise,but 
more complex, expressions which appear in Eqs. (3-23) and (3-24). The effect of 
these simplifications has heretofore remained unevaluated for finite-length cavities. 

'The formidable mathematical problem here is not amenable to a closed-form 
solution, and it was necessary to use numerical means. The calculation scheme 
was essentially a direct iteration, the steps of which may be outlined as follows: 
First, values of the parameters Lid and e were chosen in order to specify the prob¬ 
lem. Next, a trial solution for €„(*') was guessed over the range 0 x' Lid. 
As will be described later, it was possible to make very favorable guesses by taking 
advantage of prior solutions. These guessed values for e a (x') served as input data 
to the right side of Eq. (3-24). For a given value of radial position r', the integra¬ 
tion was carried out numerically to yield the value of e a at that r'. This could be 
repeated for each point in the range 0 *£ r' *£ 1, and in this way there was generated 
a function € a (r') corresponding to the guessed distribution € a (x'). The e a (r') and 
€ a {x') were then utilized in evaluating the integrals of Eq. (3-23). For a fixed 
x'o, numerical integration of Eq. (3-23) provided an e a corresponding to that 
x'o. By repeating the process for all x' 0 in the range 0 ^ x' 0 *£ Lid, a new func¬ 
tion € 0 (x') or € a (x' 0 ) was obtained which could be used as input data to Eq. (3-24), 
thereby beginning another cycle in the iteration. This procedure led to a steady 
convergence to the final result, and stability problems were not encountered. 








38 


ARTIFICIAL SOURCES 


"After a few solutions were carried out, it was found that, for a fixed e, the distribu¬ 
tion of ea along the cylindrical wall was not highly sensitive to the Ltd ratio. Thus 
the L/d = 00 results from Reference [9] were utilized as the first guess for e a (x ) for 
the L/d — 4 case; the converged results for L/d = 4 were utilized for starting values 
for €„(*) for the L/d = 3 case; etc. This ordering of the solutions — from high to 
low L/d values — helped somewhat to ease what still remained a formidable com¬ 
putational task. 

"The actual numerical calculations were performed on an IBM 704 electronic 
digital computer. Solutions were carried out for L/d — 4, 3, 2, 1, 0.5, and 0.25 for 
e = 0.9, 0.75, and 0.5. For this emissivity range, it was found unnecessary to extend 
the calculations to lengths beyond L/d = 4, since nearly blackbody conditions 
already prevailed over the end disk.” 

The article presents results only for cylindrical cavities, shapes which are not of the 
greatest practical importance. 

3.2. NBS Standards 

The National Bureau of Standards does not have an official standard blackbody which 
it uses for comparison tests. It does, however, check thermocouples and other tem¬ 
perature-measuring devices. As the section on theory (Sec. 3.1.3) shows, it is easy 
to design a blackbody which gives high emissivity, but the temperature must be known 
accurately. The Bureau does prepare and issue secondary standards in the form of 
total radiance and irradiance lamps. Instructions for the use of these sources are 
given below: 

3.2.1. “Instructions for Using the Total Radiation Standards 
(Revised February 24, I960)” 

"These instructions cover the use of our small carbon filament standards of 
thermal radiation. The lamps employed (about 50 watts) have been seasoned, 
marked for orientation, and calibrated for density of radiant flux at a fixed distance 
in a specified direction from the lamp. Suitable markings (a line on one side and 
a line through a circle on the opposite side) have been etched into the glass of the 
lamp bulbs to assist the operator in setting up the lamps relative to the radiometer 
in a position identical to that under which the standards were calibrated. 

"In operation the lamp is to be screwed into an ordinary metal socket that is held 
upright by a support, which cannot reflect light into the radiometer. The lamp is 
to be oriented so that the etched line on one side and the line and circle on the 
opposite side are in line with the radiometer. The circle on the lamp is to be 
situated away from the radiometer and on the horizontal axis through the radiom¬ 
eter. The entire lamp bulb is to be exposed to the radiometer. Sufficient time 
must be given (say five minutes) for the glass base, which supports the filament, 
to become thoroughly warmed, otherwise errors will be introduced into the radiant 
energy measurements. The distance of the lamp is measured from the radiom¬ 
eter to the center (glass tip if present, the etched mark, or other special mark 
noted in the 'Report’ on a particular standard lamp) of the lamp bulb. 

"A black cloth, of about 1 meter by 1 meter edge, should be placed about 1 meter 
to the rear of the lamp. An opaque shield about 1 m by 1 m, having an opening 
about 10 cm wide and 15 cm high, is placed at a distance of about 25 cm in front 
of the lamp. To screen this opening, a shutter, about 20 by 20 cm, is placed be¬ 
tween this shield and the lamp. Facing the opening in the shield, the radiometer 
is placed at a distance of 2 m from the lamp. In this manner constant extraneous 


NBS STANDARDS 


39 


conditions are maintained between the lamp and the radiometer when the shutter 
is opened and closed. The shield and shutter may be made of air-separated sheets 
of cardboard, asbestos board, metal or other suitable materials. 

"Before the lamp is lighted, the shutter should be opened and closed to determine 
the amount of stray thermal radiation falling upon the radiometer. This test 
may be applied at any time provided the lamp has been given sufficient time to 
come to room temperature. The wall and screen to the rear of the lamp may be 
cooler than the shutter, which will cause a negative deflection. The correction 
to the observed lamp deflection is, in that case, positive. It is desirable to make 
the calibration in a dimly lighted room to avoid errors from sunlight which is 
continually varying with cloudiness, thus varying stray radiant energy within 
the room as well as the temperature of the walls, and also causing air currents 
near the radiometer. 

"The values of radiant flux of this Bureau’s primary standard of radiation are 
based upon direct measurements in absolute value; and upon a direct comparison 
with a blackbody using the Stefan-Boltzmann constant of total radiation of a 
blackbody 8 = 5.7 X 10~ 12 watt per cm 2 , as described in NBS Bulletin No. 227, 
Vol. 11, p. 87, 1914 [101, and NBS, Journal of Research No. 578, Vol. 11, p. 79, 
1933 [111. The absolute values of the primary standard are accurate to about 
1 percent. The values of the secondary standards, compared with the primary 
standard, are in agreement within about 0.5 percent. The overall accuracy in 
the use of these standards is somewhat dependent upon the conditions of tem¬ 
perature and humidity existing during their operation. Highest accuracy will 
appear for a room temperature of about 25° C and a relative humidity of about 
60 percent near which the original calibrations were made. For extreme condi¬ 
tions of temperature and of relative humidity, corrections may be required, but 
can usually be neglected (see NBS Journal of Research , Vol. 53, p. 211, 1954) [12]. 

"The best results are to be obtained by operating the lamp on 0.30 to 0.35 ampere. 
This is the current through the lamp. If the measurement of current be made 
when a voltmeter is in the circuit with the lamp, then a correction may have to 
be made to the observed current. 

"The measurement of current through the lamp is, of course, sufficient to deter¬ 
mine the radiant flux, the voltage being useful mainly to determine whether the 
lamp has remained constant. 

"To conserve the calibration, which gradually changes with use, these lamps 
should be kept as reference standards only, and other lamps used as working 
standards in all cases where extensive radiometric comparisons are made. 

"These instructions and standards of radiation apply to radiometers used in air. 
If a window be used on the radiometer, as for example in a vacuum radiometer, 
then a correction has to be made for the radiant flux absorbed by the window, 
for the particular lamp used as a standard and for the source measured. This 
absorption is a function of the temperature of the lamp filament. For example, 
it was found that for a glass or quartz window about 1.5 mm in thickness, the 
transmission amounted to about 83 percent when a certain standard lamp was 
operated on 0.35 amp., and increased to 84 percent when the lamp was operated 
on 0.40 amp. Using a fluorite window, the transmission is higher (about 91.5 
percent) and varies but little with the current ordinarily used in the lamp. For 
example, using a certain standard lamp, the transmission through a fluorite 
window varied from 91.0 percent on 0.25 amp. to 92 percent on 0.4 amp., with 
an average value of 91.6 percent on 0.35 amp. 


40 


ARTIFICIAL SOURCES 


"The transmission of the window varies also with the spectral quality of the 
radiant flux emitted by the source under investigation. This must also be taken 
into consideration. 

"The thermal radiation sensitivity of a surface thermopile varies with the degree 
of evacuation; when highly evacuated this sensitivity may be several times as 
great as in air. Since at low air pressures the sensitivity is variable with the 
pressure, great care must be taken to test the thermopiles sensitivity under the 
exact conditions existing during its use. 

"The same area of the radiometer receiver should be exposed to the standard of 
radiation as is used in the measurements of the unknown source.” 

3.2.2. “Instructions for Using the NBS Standards of Spectral Irradiance 
(May 31, 1963)” 

"These instructions cover the use of tungsten-filament quartz-iodine lamps issued 
as standards of spectral irradiance for the wavelength range of 0.25 to 2.6 microns. 
The lamps employed are commercial G.E. type 6.6A/T4Q/lCL-200-watt lamps 
having a tungsten coiled-coil filament enclosed in a small (1/2 inch X 2 inches) 
quartz envelope containing a small amount of iodine. 

"The spectral radiant intensity of the entire lamp as mounted in the manner 
prescribed below is measured and reported. The spectral irradiance from these 
lamps is based upon the spectral radiance of a blackbody as defined by Planck’s 
equation and has been determined through comparison of a group of quartz iodine 
lamps with (1) the NBS standards of spectral radiance, (2) the NBS standards of 
luminous intensity, and (3) the NBS standards of total irradiance. 

"The lamp is mounted vertically with the NBS-numbered end of the lamp down 
and the tip away from the detector. Measurements of distance (from lamp fila¬ 
ment) are made along a horizontal axis passing through the center of the lamp 
filament. The correct vertical position is determined by setting the centers of 
the upper and lower seals along a plumb line as viewed from one side of the lamp. 
The plane of the front surface of the lower press seal is set to contain the horizontal 
perpendicular to the line connecting the lamp filament center and detector. 

"The lamp is mounted in the supplied holder which is constructed in such a man¬ 
ner as to reflect a negligible amount of radiant flux in the direction of the radiom¬ 
eter or spectrometer slit. A black shield should be placed at a distance of about 
3 feet to the rear of the lamp to intercept stray radiant flux along the radiometric 
axis and adequate shielding should be provided to intercept stray flux from other 
directions. 

"If there is excessive water vapor in the laboratory atmosphere, errors may result 
at the wavelengths of water-vapor absorption bands. In the original calibrations 
the comparisons of the lamps with the other NBS standards were made in such a 
manner that the effect of water-vapor absorption was eliminated. 

"Values of spectral irradiance for these lamps are tabulated as a function of wave¬ 
length in microwatts per (square centimeter-nanometer) at a distance of 43 centi¬ 
meters from center of lamp to receiver. Values of spectral irradiance for wave¬ 
length intervals other than one nanometer, say x nanometers, may be found by 
multiplying the tabulated values by x. 

Use of the Standards of Spectral Irradiance 

"These standards require no auxiliary optics. If any are employed proper correc¬ 
tion must be made for their optical characteristics. The lamp is simply placed 
at a measured distance from the detector or spectrometer slit. If a distance other 


NBS STANDARDS 


41 


than 43 centimeters is used, the inverse-square law may be used to calculate the 
spectral irradiance (the inverse-square law should not, however, be used for dis¬ 
tances shorter than about 40 centimeters). 

In measurements wherein two sources (a standard source and a test source) are 
being compared by the direct substitution method (slit widths kept unchanged, 
use of the same detector) no knowledge of the spectral transmittance of the spec¬ 
trometer, nor of the spectral sensitivity of the detector is required. It is necessary, 
however, to make sure that the entrance slit of the spectrometer is fully and uni¬ 
formly filled with radiant flux both from the standard and from the test source; 
and if at any one wavelength the detector response for the standard is significantly 
different from that for the test source, the deviation from linearity of response of 
the detector must be evaluated and taken into account. Furthermore, if the stand¬ 
ard and test source differ in geometrical shape, it must be ascertained that the 
instrument transmittance and detector response are not adversely affected thereby. 
Many detectors are highly variable in sensitivity over their surface area and may 
require diffusion of radiant flux over their surface to insure accurate radiant 
energy evaluations. 

"All calibrations were made by the use of alternating current and it is recom¬ 
mended that they be so used in service. To reduce the line voltage a 10-ampere 
variable autotransformer may be employed for coarse control. For fine control a 
second (5-ampere) variable autotransformer may be used to power a radio-filament 
transformer whose secondary (2.5-5 volt) winding is wired in series with the 
primary of the 10-ampere transformer. It was found that this method is very 
effective for accurate control of the 6.50-ampere current. 

"These standards of spectral irradiance are expensive laboratory equipment 
and it is suggested that they be operated sparingly and with care in order to prolong 
their useful life. They should be turned on and off at reduced current and great 
care should be taken so that at no time will the current appreciably exceed 6.50 
amperes. It is recommended that for general use, working standards be prepared 
by calibrating them relative to the laboratory standard supplied by NBS. 

"These lamps operate at high temperature such that the quartz envelope is above 
the flammable point of organic materials. They may thus cause fires, and also 
the burning of lint, etc. on the enveloper which may result in optical damage to 
its surface. In no case should the fingers come into contact with the quartz en¬ 
velope, either hot or cold, as the resulting fingerprints will burn into its surface 
during lamp operation. 

3.2.3. “Instructions for Using Standards of Spectral Radiance 
(Revised February 21, 1961)” 

General Discussion 

"These instructions cover the use of ribbon-filament lamps issued as standards 
of spectral radiance for the wavelength ranges of 0.25 to 0.75, 0.5 to 2.5, and 0.25 
to 2.5 microns. The lamps employed are commercial G.E. Type 30A/T24/7 lamps 
having a tungsten ribbon filament (SR-8A type) centered about 8 to 10 cm behind 
a fused silica window 3 cm in diameter. 

"The portion of the filament whose spectral radiance has been determined is the 
central portion visible through the fused-silica window. This determination was 
made by direct substitution of the lamps for working standard lamps which had in 
turn been calibrated by this substitution method relative to blackbodies (operated 
from 1400 to 2300°K) through the use of a double quartz prism spectroradiometer 
and associated electronic equipment. 


42 


ARTIFICIAL SOURCES 


"In operation the lamp is mounted vertically and the beam of radiant flux with a 
horizontal axis passing through the center of the filament is measured. In the 
original determinations no portion of the beam measured departed from this axis 
by more than 2.5 degrees. Hence, if an aperture subtending a larger angle is 
required in any application of these standards of spectral radiance, it should be 
ascertained that the irradiance is constant over the whole aperture. 

"If there is excessive water vapor in the laboratory atmosphere errors may result 
at the wavelengths of water-vapor absorption bands. In the original calibrations 
the comparisons of the lamps with the blackbodies were made at the same distance 
and in such manner that the effect of water-vapor absorption canceled out. 

"In the calibrations wherein the blackbody was heated within a wirewound fur¬ 
nace and temperatures around 1400° K were reached, the temperatures were meas¬ 
ured with Pt-Pt 10% Rh thermocouples. Tests using couples placed at various 
positions and observations with an optical pyrometer indicated closely uniform tem¬ 
peratures within the blackbody enclosure. A ratio of blackbody opening to total 
internal surface area equal to approximately 0.003 and an internal surface re¬ 
flectance less than 0.10 indicate an emissivity of 0.999 or higher. 

"For the blackbody temperatures above 1400°K a graphite enclosure heated by a 
radio-frequency generator was employed and the temperatures were measured 
by an optical pyrometer. The physical characteristics of this blackbody indicated 
an emissivity approximating 0.996. 

"The spectral radiance of the blackbody is based upon the Planck radiation law in 
which the constants, based upon the most recent atomic and other information, 
are set down as follows: 

Ci = 1.19088 X 10 “ 12 watt cm 2 per steradian 

c 2 = 1.4380 cm degree K 

"Values of spectral radiance for these lamps are tabulated as a function of wave¬ 
length in microwatts per (steradian-millimicron-square millimeter of filament). 
Values of spectral radiance for slit-widths other than one millimicron, say x milli¬ 
microns, where x is less than 100 , may be found by multiplying the tabulated 
values by x. 

Use of the Standards of Spectral Radiance 

"It is suggested that the auxiliary optics employed with these standards be com¬ 
posed of two units: namely, a plane mirror and a spherical mirror (each aluminized 
on the front surface). If the spherical mirror is placed at a distance from the 
lamp filament equal to its radius of curvature, and the plane mirror set about 
1/3 to 2/5 this distance from the spherical mirror, facing it and so placed that 
the angle between incident and reflected beams is 10 ° or less, a good image of 
the filament itself may be focused upon the spectrometer slit. Little distortion 
of the filament image occurs provided precise optical surfaces are employed and 
angles between incident and reflected beams are kept to less than 10°. Various 
optical arrangements may be employed. 

"The solid-angular aperture of the auxiliary optics should be smaller than the 
solid-angular aperture of the spectrometer employed so that no loss of radiant 
energy will result through over-filling the spectrometer optics. 

"The spectral radiant flux, P\, in microwatts per millimicron, which enters the 
spectrometer slit is computed from the formula: 


Px = RkN k sA/D 2 


NBS STANDARDS 


43 

where R \ is the spectral reflectance of the combination of mirrors used, N\ is the 
reported spectral radiance of the standard, s is the area of the spectrometer slit 
in mm 2 , A is the area of the limiting auxiliary optic, and D is the distance of this 
optic from the slit. 

No diaphragm or other shielding is required in the use of these standards, except 
for a shield to prevent radiant energy from the lamp from entering the spectrom¬ 
eter directly without first falling on the concave mirror. An image of the filament 
should be focused upon the spectrometer slit, and only the energy by which this 
image is formed should enter the slit. 

"In order to calibrate a spectroradiometer with one of these standards of spectral 
radiance, a knowledge of the spectral reflectance of the mirror surfaces is required. 

A good aluminized surface should have a spectral reflectance considerably above 
0.87 throughout the spectral region of 0.5 to 2.6 microns, increasing slightly with 
wavelength except possibly for a slight dip near 0.80 micron. In practice the proper 
reflectance losses can best be determined through the use of a third (similar) 
mirror (a second plane mirror) which may be temporarily incorporated into the 
optical set-up from time to time. 

"In measurements wherein two sources (a standard source and a test source) are 
being compared by the direct substitution method (use of the same auxiliary optics, 
slit-widths, areas and detector at any one wavelength), no knowledge of the spectral 
reflectance of the auxiliary mirrors, nor of the spectral transmittance of the spec¬ 
trometer, nor of the spectral sensitivity of the detector is required. It is necessary, 
however, to make sure that the entrance slit of the spectrometer is fully and uni¬ 
formly filled with radiant flux both from the standard and from the test source; 
and, if at any one wavelength the detector response for the standard is significantly 
different from that for the test source, the deviation from linearity of response 
of the detector must be evaluated and taken into account. 

"Operation of these standards should be on alternating current to obviate fila¬ 
ment-crystallizing effects that occur when the operation is on direct current. The 
filaments are massive and "iron out” all effects of the normal fluctuations present 
in a commercial ac supply. All calibrations were made by means of alternating 
current. To reduce line voltage a step-down transformer (1 kva) having a ratio 
of 10 to 1 or a 50-ampere variable transformer may be employed. Then to give 
fine control a second variable transformer (10-ampere capacity) is wired into the 
circuit to control the input of the heavy duty transformer. For still finer control 
a third variable transformer may be employed with a radio-filament transformer 
to add (or subtract) voltage fed into the step-down transformer. It was found 
that this method is very effective for accurate control of large lamp currents. The 
heavy duty (1 kva) step-down transformer is preferred to the 50-ampere variable 
transformer since the latter is subject to contact damage when operated for long 
intervals of time at high currents. 

"The lamp standards of spectral radiance are expensive laboratory equipment 
and it is suggested that they be operated sparingly and with care in order to pro¬ 
long their useful life. This precaution applies especially to the standards cali¬ 
brated in the short-wave region and operated at 35 amperes. They should be 
turned on and off slowly and only for short intervals should they ever be operated 
at or above 30 amperes, and then only to calibrate a similar lamp as a working 
standard. In general even at lower currents a working standard should be pre¬ 
pared and used except for purposes of checking the operation of such a working 
standard.” 


44 


ARTIFICIAL SOURCES 


3.2.4. Other NBS Sources. Figure 3-3 shows a blackbody furnace used at the 
National Bureau of Standards as a comparison standard in emissivity determinations 
[13]. The core, made of Inconel, has an outside diameter of 3 cm and a cylindrical 
cavity 2.06 cm in diameter and 7 cm deep with inner walls roughened by a very fine 
tap and treated to produce an opaque, oxidized layer. The furnace is heated by an 
80% platinum-20% rhodium resistance element surrounding the core. The core and 
heating element are covered with thermal insulation 3.8 cm thick and are mounted 
axially in a water-cooled tube of 11.4 cm outside diameter. Power input to the furnace 
is adjusted manually by an autotransformer. The temperature of the furnace is 
measured by a thermocouple inserted in a hole drilled from the back of the core to 
within 0.08 cm of the inner surface of the core at the rear of the cavity. The emissivity 
of the inner surface is greater than 0.90 and that of the furnace was computed to be 
greater than 0.99 in the temperature range for which it was designed, 810° to 1360° K. 


Cooling Jacket. Use Tube 

or Wire to Channel Flow. Cooling Coils 



Fig. 3-3. NBS blackbody furnace. 


Figures 3-4 and 3-5 show two other blackbody cavities designed by NBS. They are 
probably the closest approximation to blackbody radiators in existence. These cavities 
are primarily used in optical pyrometry as gold-pcint blackbodies to determine the 
International Practical Temperature Scale (IPTS) above 1336°K [14]. 

The vertical blackbody assembly shown in Fig. 3-4 is heated in a wire-wound muffle 
furnace or in the coils of an rf generator. The estimated emittance of this blackbody 
is 0.999, assuming the walls are at a uniform temperature. The horizontal blackbody 
shown in Fig. 3-5 has three independently controlled heater windings that are embedded 
longitudinally in cylindrical graphite muffles. 

The graphite, having high thermal conductivity, tends to reduce longitudinal tem¬ 
perature gradients. The power inputs to the two end windings are adjusted to maintain 
the end sections at a specified temperature, i.e., the IPTS [15], as determined by two 
thermocouples positioned near the inner surface of these sections. The center winding 
is used to control the temperatures of the gold during gold-point calibrations. 

Another high-temperature blackbody, used at NBS for accurate spectral calibrations 
of monochromators and spectrographs, has been used up to temperatures of 3273° K. 
It consists of a graphite cylindrical tube resistively heated in an argon atmosphere 
and surrounded by a number of graphite radiation shields. The tube is about 200 mm 













































































NBS STANDARDS 


45 


11 



Fig. 3-4. NBS vertical blackbody. 


I«V<*1 Quartz Wool 

mm Gold 

HH Graphite 

Silocel Powder 


Depth of Cavity _ 

Radius of Cavity Opening 


= 72 


Copper Cooling Coils 


Quartz Cylinders 


Argon 



Alumina Cone 


18 cm 


/ y 

^ Metal Cover 


High Temperature 
Cement 


Nichrome Windings 
in Alumina Tubes 


Cavity Brace 


Fig. 3-5. NBS blackbody and furnace. 

































































































































46 


ARTIFICIAL SOURCES 


long, has a wall thickness of about 3 mm, and an inside diameter of about 9 mm. A 
small hole in the center of the tube and correspondingly larger holes in the shields 
permit the radiation to exit. A current of about 800 amp is required to reach a tem¬ 
perature of 3073° K. The tube can be used for about 50 hours at this temperature and 
the radiance can be stabilized for several hours to better than 1 percent by automat¬ 
ically controlling the current in the tube [14]. 

3.3. Laboratory and Field Sources 

3.3.1. A High-Temperature Source [16]. "The high temperature element of 
the source is a graphite tube, with a small slit in one wall, heated directly with an 
alternating current. The arrangement is shown in cross section in Fig. 3-6. The 
graphite tube is rigidly supported at the upper end, and electrical connection is 
provided by means of a standard compression tube fitting modified by slightly 
enlarging its bore. At the other end, electrical connection and linear motion to 
relieve thermal stress are accomplished by a piston and bellows arrangement. 
The piston itself acts as the nut for another modified tube fitting; electrical insula¬ 
tion between the piston and the cylinder wall is produced by two O-rings, which 


Section A-A 



A - Graphite Plug 
B - Modified Tube Fitting 
C - Brass Bushing 
D - "O" Ring 
E - Graphite Tube 
F - .03” X .25" Slit 
G - Argon Inlet 
H - Water Inlet 
I - Water Outlet 
J - Piston 
K - Bellows 
L - Lucite Bushing 


Fig. 3-6. Blackbody of Simmons, DeBell, and Anderson. 



































































LABORATORY AND FIELD SOURCES 


47 


further serve as a fluid barrier. The bellows is mounted on a bushing of insulating 
material; electrical connection is made both to the central screw and the body 
wall. Both electrodes are cooled by water flow. 

"The heated element was made from National Carbon Company type AGR graph¬ 
ite as a tube of 0.250-in. O.D. with a 0.050-in. wall; a slit of 0.25 X 0.03 in. was 
milled through one wall at approximately halfway between the points of first 
thermal or electrical contact. Erosion and oxidation of the hot graphite is reduced 
to a negligible rate by a flush and a slight flow of dry argon, which escapes through 
the body opening in front of the slit. The ends of the graphite tube were copper- 
plated to provide better contact. 

"The power supply consists of two 20-amp toroidal autotransformers, one of which 
was converted into a conventional step-down transformer by removing the slider 
and adding a secondary winding of a few turns. Connection between this unit 
and the source body was made by means of a standard type of single-conductor 
insulated welding cable enclosed in 3/4-in. brass pipe which was connected to the 
base of the body. By this means, concentricity of the high current part of the 
electrical system is maintained; magnetic fields which would create spurious sig¬ 
nals in nearby measurement circuits are thereby minimized. 

"The current through the graphite and hence its temperature is controlled by 
manual adjustment of the unmodified autotransformer, the output of which is 
connected to the primary of the modified unit. The secondary current of the 
latter, i.e., the current through the graphite tube, is measured using a 0-5 amp ac 
ammeter with a 40/1 current transformer. The time constant of the graphite 
tube is of the order of 10 sec so that no 60-cycle ac ripple is detectable in the radiant 
output. Constancy in source temperature is maintained by close control of line 
voltage with an electronic regulator, and by careful regulation of the argon flush. 

"Calibrations were made with an optical pyrometer which in turn was checked 
against an NBS-certified tungsten ribbon lamp. The blackness of the source was 
not ascertained by experiment. However, an emissivity of 0.975 was computed 
assuming a uniform wall temperature, diffuse reflection from the inner wall, and 
an average surface emissivity of 0.85 for the graphite. (Source emissivities 
closer to unity can of course be obtained by reducing the ratio of the slit width 
to the internal tube diameter.) 

"The highest temperature of operation is at present limited by the power supply 
to about 2400° K at which approximately 2 kw are being dissipated. During 
repetitive operation over periods of a few hours, the calibration has been found 
to be very stable following the first few heating and cooling cycles, during which 
a slight amount of further graphitization is presumed to have occurred. On 
occasion, over longer periods of many hours, appreciable drifts in the calibration 
have been observed. These shifts were found to be due to a gradual ablation of 
the graphite at a rate primarily related to the temperature of the graphite and 
the dryness of the argon. In the intermittent service for which the source was 
designed, however, the calibrations of the several units in use at Rocketdyne ex¬ 
hibit only very slight changes, which are monitored by occasional recalibrations.” 

3.3.2. Low-Temperature Large-Area Sources. Large-area thermal radiation 
sources used for comparison with field sources have been constructed [17]. One is 
an ambient-temperature conical field source, and another is a temperature-controllable 
conical field source. The ambient-temperature source consists of a simple sheet- 
metal cone coated internally with Sicon-black enamel. This source quickly assumes 
the ambient air temperature and is not temperature controlled. A thermometer 


48 


ARTIFICIAL SOURCES 


mounted so as to insure good thermal contact does the monitoring. The temperature 
distribution on the inner surface of the cone under normal conditions was determined 
with a surface pyrometer to be uniform within 0.5° C or less. 

The temperature-controllable source consists of a similar cone with a water jacket. 
Provision is made for circulation and heating of the water. The impeller mounted 
near the apex and a stirrer blade are arranged such that a temperature uniformity of 
1°K is maintained from the rear to the front of the cone as well as from top to bottom 
of the front portion of the source at an average temperature of 323° K. 

Although no accurate experimental determinations have been made of the emissivity 
of these conical field sources, calculations have been made using the methods of DeVos 
[1] as applied to a conical source by Edwards [2]. The resulting value is 0.99. 

Other field sources have been constructed commercially for use in comparing the 
radiance of large area targets. One in particular that has been used in the past is a 
1 ft 2 flat metal surface covered with Zapon paint and temperature controlled, by means 
of a winding on the back, to a maximum temperature of about 250° C. The use of this 
"blackbody,” manufactured by Barnes, and of most flat ones, is restricted because of 
the nonnegligible gradients of temperatures across the surface. 

3.3.3. Nernst Glower. The Nemst glower is a hollow rod or filament about 3 cm 
long and 1 mm in diameter made from Zr0 2 (zirconia) and Y 2 0 3 (yttria) mixed with 
Ce0 2 (ceria) or Th0 2 (thoria). It is heated by an electric current from an approximately 
180-w 50-v power supply. When used as a spectrometry source, the Nernst glower is 
intended to match entry slits about 12 mm high. The Nernst glower is generally 
operated at temperatures between 1500° and 2000°K, and has an expected life of 
about 200 to more than 1000 hours. Platinum wires are used as electrodes; therefore 
water cooling of end connections is not required. The connections are critical; con¬ 
struction details are given in Rev. Sci. Inst., Vol. 11, p. 429 (1940) [18] and Phil. Mag., 
Vol. No. 50, vi, p. 263 (1925) [19]. Because of its negative temperature coefficient of 
resistance, the Nernst glower must be heated to start and will continue to heat up 
until it burns out unless it is connected in series with a resistance to limit the current 
(ballasted). 

The Nemst glower can be used for wavelengths up to about 30 /a; beyond 15 /x its 
emissivity decreases and its performance is comparable to that of the Globar or the 
Welsbach mantle. Low mechanical strength, small size, and susceptibility to tem¬ 
perature variations caused by air currents are disadvantages of the Nernst glower [20, 
21 ]. 

3.3.4. Globar. The Globar is a rod of bonded silicon carbide. In its most common 
form it is about 5 cm long and 5 mm in diameter. It is heated electrically from a 
power supply of approximately 180 w at 50 v, and has a lifetime of about 250 hr when 
operated at 1500°K in air. It can be operated at temperatures up to 2200° K by a tech¬ 
nique described by Strong [21]. Silver electrodes are used, and the ends of the Globar 
are coated with silver. A water-cooled jacket is used to prevent overheating of the 
end connections. 

The Globar closely approximates a graybody, its emissivity being about 80% from 
4 to 15 ix as shown in Fig. 3-7. Globar rods are available in large sizes for use as ex¬ 
tended sources and for heating elements in furnaces and ovens operating up to about 
1600°K [20], 

3.3.5. Welsbach Mantle. The Welsbach mantle is a gauze similar to that used in 
gasoline lamps and lanterns. The gauze is impregnated with Th0 2 to which 0.75 
to 2.5 percent Ce0 2 is added. The gauze, or mantle, is heated either by burning gas 


LABORATORY AND FIELD SOURCES 


49 


or an electric current (or both). The emissivity of the Welsbach mantle is relatively 
low up to about 6 /lx, but between 10 and 100 /x it approximates a blackbody and is 
comparable to most other sources. At wavelengths exceeding 150 /lx the emissivity 
falls off to the point where it is no longer usable. The Welsbach mantle can be operated 
at temperatures up to about 2400° K [20,21]. 



24 6 8 10 12 14 16 


WAVELENGTH [ ^) 

Fig. 3-7. Spectral output of Globar [22], 


3.3.6. Carbon Arc. High source brightness is attained with the carbon arc, of 
which there are two main types: low intensity and high intensity. In the low-intensity 
arc, the radiation comes mainly from solid incandescent carbon in the shallow crater 
created on the face of the positive electrode. The temperature in this region is ap¬ 
proximately 3900° K. In high-intensity arcs, a deeply formed crater appears in the 
specially cored positive electrode. This crater is characterized by a higher tempera¬ 
ture and a higher current density than that in the low-intensity arc. Emission from 
vaporized core materials adds a dense visible line spectrum to the continuous spectrum 
emitted by the crater. The color temperature of the high-intensity arc can be as low 
as about 5000° K to as high as about 9000° K depending upon core material and current. 

The electrode size, and therefore the current rating of a low-intensity arc, is optional 
since all low-intensity arcs have essentially the same positive crater temperature. 
The larger the electrode size, the larger the crater and the greater the ease of illuminat¬ 
ing a given area, such as a spectrometer entrance slit. Typical carbon electrodes 
are 12 mm for the positive and 8 mm for the negative. A ballast resistor provides 
current stability with the negative coefficient of resistance of carbon. 

The carbon arc is an excellent source in the wavelength range of 10 to 100 /lx; beyond 
100 /x a high-pressure mercury arc is better. Best source stability (better than ±3%) 
is achieved by masking all but the crater; this is the most uniform part of the source. 
Details of operation of these sources are given in [23] and [24]. 

3.3.7. Tungsten Filaments. The tungsten-filament lamp, which operates at about 
2800°K, is a source of high brightness. Because its glass envelope is not transparent 




50 


ARTIFICIAL SOURCES 


beyond about 3 p., the tungsten-filament lamp, as normally used, is limited to about 
this wavelength. If, however, a tungsten filament is mounted in an enclosure behind 
a suitable infrared window, its useful range is limited only by the long-wavelength 
cutoff of the window. The signal from a tungsten filament mounted behind an alkali 
halide window was found [25] to be about one-half that of a carbon arc at all wavelengths 
from 2 to 14 /x. The ratio of the tungsten lamp emittance to that of a Globar is shown 
in Fig. 3-8 [25]. 


o 

o 

o 

o\ 

CN 


c 

4 ) 

e 


CO 





WAVELENGTH [ fl) 


Fig. 3-8. Ratio of the output of a 
tungsten lamp at 2900° K to that of 
a Globar at 1330° K (replotted from 
[25]). 


Another type of tungsten lamp called a strip- or ribbon-filament lamp is often used 
by NBS for spectral radiance calibrations of monochromators or spectrographs and as 
a comparison standard. 

The G.E. Type 30A/6V/T24 tungsten strip lamp has a glass envelope about 75 mm 
in diameter and 300 mm long and a tungsten ribbon filament about 0.075 mm thick, 
3 mm wide, and 50 mm long. The envelope contains argon at a pressure of about 
0.3 atm at room temperature. This lamp requires a current of about 14 amp for a 
brightness temperature of 1273°K and about 45 amp for 2573°K. The change of 
brightness temperature with current varies from about 30 to 100°/amp from 1073° 
to 2573°K. Direct current is usually used so that standard potentiometric methods 
can be employed. 

Tungsten lamps are commercially available from the General Electric Co. in the 
United States, the General Electric Co., Ltd., in England, and Phillips Lamp Works 
in the Netherlands. 

3.3.8. Mercury Arcs. The chief laboratory source of far-infrared radiation is 
a commercially available high-pressure mercury arc operating in a fused-silica en¬ 
velope. This source can be used for wavelengths from about 50 to 1400 fx. The 
high pressure (from 1 to 100 atm) of the arc broadens the discrete line spectrum into 
a continuous series of broad peaks. As the vapor pressure is increased, more of the 
emitted radiation occurs at longer wavelengths [21]. 




COMMERCIAL CAVITY-TYPE SOURCES 


51 


Figure 3-9 shows the ratio of the intensity of a quartz-mercury lamp compared 
to a Globar source. It can be seen that the energy of the mercury lamp is about six 
times greater at 200 /x and about three times greater at 100 fi. Between 20 and 50 
cm -1 very little energy is emitted by the Globar. The quartz-mercury lamp used to ob¬ 
tain the curve was designated HPK 125 W and was made by Phillips’ Lamp Works in the 
Netherlands. It had a working pressure of 3 atm. The lamp was operated at 135 v 
dc and a current of 0.98 amp. The Globar source was the usual type for commer¬ 
cial infrared spectrometers and was operated at 1200° K [26]. Other types of discharge 
lamps, notably H, Hg, and Xe lamps, can be used also as infrared sources. 



Fig. 3-9. Ratio of quartz-mercury lamp 
output to that of a Globar (replotted from 
[26]). 


3.3.9. Zirconium Point Sources. An interesting type of arc, useful when a very 
small source of light is needed, is the so-called concentrated-arc lamp. The cathode 
consists of a small metal tube packed with zirconium oxide and the anode consists 
of a metal plate containing a hole slightly larger than the end of the cathode. Tungsten, 
tantalum, or molybdenum, because of their high melting points, are used for the metal 
parts. These are sealed in a glass bulb which is filled with an inert gas like argon 
to a pressure of nearly 1 atm. The arc runs between the (fused) surface of the zir¬ 
conium oxide and the surrounding anode. The tip of the cathode is heated by ion 
bombardment to 2700°C or higher, giving it a surface brightness almost equal to 
that of the carbon arc. The light is observed through the hole in the anode. Lamps 
of this type can be made in which the source diameter is as small as 0.007 cm. These 
lamps are listed on pages 336 and 337 of Cenco Catalog J200. (They are probably 
also available elsewhere.) They come in powers rated from 2 to 300 w. These lamps 
must be ballasted; the Sylvania lamps come with operating instructions. 

3.4. Commercial Cavity-Type Sources 

Data on cavity-type blackbody sources currently available from various manu¬ 
facturers are given in Table 3-3. Some can be obtained with aperture plates and 
other fittings. 





52 


ARTIFICIAL SOURCES 


Table 3-3. 

Currently Available 

Cavity-Type Blackbody Sources 


Mfr.;* 
Model No. 

Type of 

T emperature 

Field 

Aperture 

Diameter 

Ernissivity 

Warm-Up 

Max. 

Input 

Power 

(w) 

Temp. 

Radiation 

Source 

Range 

(°C) 

Accuracy 

(°C) 

of View 
(degrees) 

Time (min); 
Temp. (°C) 

Measuring 

Element 

Barnes 

14° 

60-230 

±1" 

20 

0.50 in. 

0.99 

30 min 

25 

Th 

11-100 

conical 

cavity 




12.7 mm 

±1% 

230° C 



Barnes 

14° 

0-230 

±1" 

20 

0.625 in. 

0.99 

30 

100 

Pt 

11-101 

conical 

cavity 




16.9 mm 

±1% 

230 



Barnes 

28° 

200-600 

±3" 

20 

0.50 in. 

0.99 

120 

400 

Pt 

11-110 

conical 

cavity 




12.7 mm 

±1% 

600 



Barnes 

14° 

200-600 

±3 n 

20 

0.015 in. 

0.95+ 

2 

55 

Pt 

11-120 

conical 

cavity 




0.38 mm 


200 



Barnes 

14° 

200-600 

±3 

20 

0.04 in. 

0.95+ 

2 

55 

Pt 

11-121 

conical 

cavity 




1 mm 


200 



Barnes 

14° 

200-1000 

±5" 

20 

0.40 in. 

0.99 

60 

800 

Pt 

11-131 

conical 

cavity 




10.2 mm 

±1% 

1000 



Perkin-Elmer 

15° 

50-600 

±1* 

20 

0.50 in. 


20 

160 

Pt 

PE 521-4 

conical 




12.7 mm 


600 



(Source) 

cavity, 









PE 521-5 

blackened 









(Controller) 

silver cone 









Perkin-Elmer 

15° 

200-600 

±2 r 

20 

0.0087 in. 


<3 

55 

Pt 

PE 521-6 

conical 




0.2 mm 


600 



(Source) 

cavity, 









PE 521-5R 

blackened 




0.015 in. 





(Controller) 

silver 




0.38 mm 






cone 




0.040 in. 
1.02 mm 





Perkin-Elmer 

15° 

200-600 

±1.5 <> 

20 

0.040 in. 





PE 521-1 

conical 




1.02 mm 





(Source) 

cavity, 









PE 521-5 

blackened 




0.015 in. 


<3 

55 

Pt 

(Controller) 

silver 




0.38 mm 


600 




cone 









ITT 

15° 

200-600 

±2 


0.33 in. 



250 

Pt 


blackened 

conical 

cavity 




8.4 mm 





ITT 

Blackened 

40-300 

±1" 


0.375 in. 




TC 


conical 

cavity 




9.5 mm 





IR Ind. 

20° 

50-710 

±1 

14 

0.0200 in. 

0.99 

30 

250 

Pt 

IRI 403 

blackened 




5.08 mm 

±0.01% 




(Source) 

stainless 









IRI 101 

steel 




0.100 in. 





(Controller) 

cone 




2.54 mm 










0.050 in. 
1.27 mm 










0.025 in. 
0.64 mm 






0.0125 in. 
0.32 mm 



COMMERCIAL CAVITY-TYPE SOURCES 


53 


Table 3-3. Currently Available Cavity-Type Blackbody Sources ( Continued) 


Mfr.;* 
Model No. 


IR Ind. 

IRI 404 
(Source) 

IRI 101 
(Controller) 


Type of 
Radiation 
Source 

20 ° 

blackened 

stainless 

steel 

cone 


T emperature Field 

Accuracy °f View 


Range 

(°C) 

50-1000 


" (degrees) 


±1 


14 


Aperture 

Diameter 


0.200 in. 
5.08 mm 

0.100 in. 
2.54 mm 


Emissivity 


0.99 

± 0 . 01 % 


Warm-Up 
Time (min); 
Temp. (°C) 

45 


Max. 

Input 

Power 

(w) 

525 


Temp. 

Measuring 

Element 

Pt 


IR Ind. 

IRI 405 
(Source) 

IRI 101 
(Controller) 


20 ° 

blackened 

stainless 

steel 

cone 


50-710 


±1 


30 


IR Ind. 

IRI 406 
(Source) 

IRI 102 
(Controller) 


Blackened 

conical 

cavity 


200-600 


30 


0.050 in. 
1.27 mm 

0.025 in. 
0.64 mm 

0.0125 in. 
0.32 mm 

0.600 in. 
15.3 mm 

0.400 in. 

10.2 mm 

0.200 in. 
5.05 mm 

0.100 in. 
2.54 mm 

0.050 in. 
1.27 mm 

0.025 in. 
0.64 mm 

0.0125 in. 
0.32 mm 

0.600 in. 

15.3 mm 

0.400 in. 
10.2 mm 

0.200 in. 
5.05 mm 

0.100 in. 
2.54 mm 

0.050 in. 
1.27 mm 

0.025 in. 
0.64 mm 


0.99 

: 0 . 01 % 


60 


525 


Pt 


0.99 

±0.01% 


60 


525 


Pt 


0.0125 in. 
0.32 mm 


0.039 in. 0.99 5 15 Pt 

0.99 mm ±0.01% 

(Source) cavity 

IRI 102 
(Controller) 


IR Ind. Blackened 200-600 ±1 10 

IRI 407 conical 



54 


ARTIFICIAL SOURCES 


Table 3-3. Currently Available Cavity-Type Blackbody Sources ( Continued) 


Mfr.;* 
Model No. 

IR Ind. 

IRI 408 
(Source) 

IRI 102 
(Controller) 


IR Ind. 

IRI 411 
(Source) 

IRI 101 
(Controller) 

IRI Ind. 

IRI 412 
(Source) 

IRI 106 
(Controller) 

IRI Ind. 

IRI 417 
(Source) 

IRI 103 
(Controller) 


IR Ind. 

IRI 420 
(Source) 

IRI 101 
(Controller) 


IR Ind. 

IRI 424 
(Source) 

IRI 101 
(Controller) 


IRI Ind. 

IRI 427 
(Source) 

IRI 102 
(Controller) 

Radiation 

Elec. 


Type of 

Temperature 

Field 

Aperture 

Diameter 


Warm-Up 

Max. 

Input 

Power 

(w) 

Temp. 

Radiation 

Source 

Range 

(°C) 

Accuracy 

(°C) 

of View 
(degrees) 

Emissivity 

Time (min); 
Temp. (°C) 

Measuring 

Element 

Blackened 

conical 

cavity 

200-600 

±1 

90 

0.100 in. 
2.54 mm 

0.050 in. 
1.27 mm 

0.99 

±0.01% 

5 

60 

Pt 





0.025 in. 
0.64 mm 









0.0125 in. 
0.32 mm 





Blackened 

conical 

cavity 

710-1700 

±1 

14 


0.99 

±0.01% 

90 

550 

Pt 

Blackened 

50-900 

±1 

90 

4.5x4.5 in. 

0.90- 

60 

800 

Pt 


conical 114.3X 0.97 

cavity 114 3 mm 


Blackened 50-1000 ±1 

conical 
cavity 


Blackened 200-1200 ±1 

conical 
cavity 


Blackened 710-1700 ±1 

conical 
cavity 


Blackened 200-600 ±1 

conical 
cavity 


18 2.0 in. 0.99 

50.8 mm ±0.01% 

1.5 in. 

38.1 mm 

1.0 in. 

25.4 mm 

14 0.200 in. 0.99 

5.08 mm ±0.01% 

0.100 in. 

2.54 mm 

0.050 in. 

1.27 mm 

0.025 in. 

0.64 mm 

0.0125 in. 

0.32 mm 

14 0.050 in. 0.99 

1.27 mm ±0.01% 

0.025 in. 

0.64 mm 

0.0125 in. 

0.32 mm 

14 0.080 in. 0.99 

2.03 mm ±0.01% 


90 1100 Pt 


45 380 Pt 


90 550 Pt 


3 15 Pt 


V-grooved 60-520 

±2 

2.125 in. 

0.99 

30 

225 TC 

metal 

block 


54.0 mm 

±0.01% 

1000° F 

(Fe- 

constantan) 



REFERENCES 55 


Table 3-3. Currently Available Cavity-Type Blackbody Sources ( Continued) 


Mfr.;* 
Model No. 

Type of 

Temperature 

Field 

Aperture 

Diameter 

Warm-Up 

Max. 

Input 

Power 

(w) 

Temp. 

Radiation 

Source 

Range 

(°C) 

Accuracy 

(°C) 

of View 
(degrees) 

Emissivity Time (min); 

Temp. (°C) 

Measuring 

Element 

Elec. 

Communi¬ 

cations 

Cylindrical 

blackbody 

cavity 

Ambient 
to 1000 
continu¬ 
ously ad¬ 
justable 

±1 


1.125 in. 
28.6 mm 

0.995 70 

1000 

700 

max. 

TC 
in air 

Eppley 

Labs. 

Blackened 

stainless 

steel 

cavity 

600-1100 

±1 


0.75 in. 
19.1 mm 

0.97 .120 approx. 

1000 

1500 

TC 

(Pt-Pt 

10% Rh) 

Williamson 

Cylindrical 

blackbody 

cavity 

Ambient 
to 65 

±1 


0.75 in. 
19.1 mm 

60 

65 

<10 

Mercury-in¬ 
glass ther¬ 
mometer 


*Barnes = Barnes Engineering Company. 

Perkin-Elmer = Perkin-Elmer Corporation. 

ITT — International Telephone and Telegraph Corporation. 

IR Ind. = Infrared Industries. 

Radiation Elec. = Radiation Electronics Company. 

Elec. Communications = Electronics Communications, Inc. 

Eppley Labs. = Eppley Laboratories, Inc. 

Williamson = Williamson Development Company. 

“Maximum error due to combined shift of calibration, ambient temperature (0° to 40°C) and line voltage 
(105 to 125 v). 

‘Temperature variations from 16° to 38°C and line-voltage variation from 105 to 125 v, 60 cycles ac. 
“Temperature variations from 20° to 40°C and line-voltage variation from 105 to 125 v, 60 cycles ac. 
d At 300° C with smaller variation at lower temperature. 


References 

1. J. C. DeVos, Physica, 20, 669 (1954). 

2. David F. Edwards, "The Emissivity of a Conical Blackbody,” 2144-105-T, The University 
of Michigan Engineering Research Institute (now Institute of Science and Technology, The 
University of Michigan), Ann Arbor (1956). 

3. Andre Gouffe, Rev. Optique, 24, 1 (1945). 

4. C. S. Williams, J. Opt. Soc. Am., 51, 564 (1961). 

5. E. M. Sparrow, L. U. Albers, and E. R. G. Eckert, J. Heat Transfer, 73 (1962). 

6. G. J. Zissis, "Precision Radiometry-Theory,” Special Topics in Infrared Technology, Engineer¬ 
ing Summer Conferences, The University of Michigan, Ann Arbor (1963). 

7. E. W. Truenfels, J. Opt. Soc. Am., 53, 1162 (1963). 

8. E. M. Sparrow, L. V. Albers, and E. R. G. Eckert, J. Heat Transfer Trans. ASME (1962). 

9. E. M. Sparrow and L. V. Albers, J. Heat Transfer Trans. ASME, Ser C, 82, 233 (1960). 

10. NBS Bulletin No. 227, 11, 87 (1914). 

11. NBS J. Research No. 578, 11, 79 (1933). 

12. NBS J. Research, 53, 211 (1954). 

13. W. N. Harrison, Joseph C. Richmond, Earle K. Plyler, Ralph Stair, and Harold K. Skramstad, 
"Standardization of Thermal Emittance Measurements,” WADC TR 59-510, National Bureau 
of Standards (1960) AD 238 918. 

14. H. J. Kostowski and R. D. Lee, "Theory and Methods of Optical Pyrometry,” NBS Monograph 
41, U. S. Department of Commerce, National Bureau of Standards. 

15. H. F. Stimson, J. Research Natl. Bur. Standards, 42, 209 (1949); J. Research Natl. Bur. Stand¬ 
ards, 65A, 139 (1961). 

16. F. Simmons, A. G. DeBell, and Q. S. Anderson, Rev. Sci. Instr., 32, 1265 (1961). 

17. A. LaRocca and G. Zissis, Rev. Sci. Instr., 30, 200 (1959). 

18. Rev. Sci. Inst., 11, 429 (1940). 

19. Phil. Mag., 50, 263 (1925). 

20. P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, Elements of Infrared Technology- 
' Generation, Transmission, and Detection, Wiley, New York (1962). 



56 


ARTIFICIAL SOURCES 


21. J. Strong, Procedures in Experimental Physics, Prentice-Hall, Inc., New York (1939). 

22. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detection and Measurement of Infra-Red 
Radiation, Oxford University Press, Oxford (1957), originally from S. Silverman, J. Opt. Soc. 
Am., 38, 989 (1948). 

23. Symposium on Searchlights, The Illuminating Engineering Society, 32 Victoria St., London 
SW1, printed by Argus Press, Ltd., Temple Ave. and Tudor St., London EC4, England. 

24. W. Finkelnberg, "The High Current Carbon Arc and Its Mechanism,” J. Appl. Phys., 20, 468 
(1949). 

25. J. H. Taylor, C. S. Rupert, and J. Strong, J. Opt. Soc. Am., 41, 626 (1951). 

26. E. K. Plyler, D. J. C. Yates, and H. A. Gebbie, J. Opt. Soc. Am., 52, 859 (1962). 


Chapter 4 
TARGETS 


C. E. Dunning 

Aerojet-General Corporation 

Fred E. Nicodemus 

Sylvania Electronic Systems 


CONTENTS 


4.1. Introduction.58 

4.2. Aerial Targets.58 

4.2.1. Symbols, Definitions, and Equations.58 

4.2.2. Jet Structure.60 

4.2.3. Exhaust Composition.62 

4.2.4. Heating During Exit.68 

4.2.5. Heating During Reentry.69 

4.3. Surface Targets.69 

4.4. Radiometric Analysis and Discussion.69 

4.4.1. Thermal Emission.70 

4.4.2. Reflection.72 

4.4.3. Factors Affecting the Temperatures of Passive Targets.72 

4.5. Measured Values of Radiant Emissivity and Reflectance.74 


57 

















4. TARGETS 


4.1. Introduction 

Although a great deal of military infrared technology does not differ greatly from 
infrared technology in general, and can be freely treated in an unclassified work such 
as this Handbook, security classification forbids such discussion in certain areas. 
The characteristics of military infrared targets is just such an area, so that this chapter 
is unavoidably incomplete. About all that can be said is that, within this severe limi¬ 
tation, an attempt has been made to include as much information as possible. 

In the first part of this chapter are data concerning rocket engines, which may be 
helpful in attempts to estimate their characteristics as infrared targets. In the second 
part radiometric principles are discussed as they apply to the radiation from a wide 
variety of targets, particularly of surface targets as they appear in the output of an 
infrared scanner. Also included are data regarding the emissivities of a number of 
common objects and surfaces. 

4.2. Aerial Targets (Rocket-Propulsion Systems) 

Aerial targets may be military objectives; they may be aircraft or rockets which are 
a threat to security, or they may be spacecraft which must rendezvous or track and 
home on a planet. The power plants can be piston engines, jets, or rockets. The 
radiation comes from the exhausts of the engines, from frictional heating of the struc¬ 
ture passing through the air, and from the air heated by very high speed objects like 
reentry bodies. 

4.2.1. Symbols, Definitions, and Equations 

A = nozzle cross-section area (in. 2 ) 
c = effective exhaust velocity (ft sec -1 ) 

Ch = Stanton number (proportional to local skin-friction coefficient, highly dependent 
on state of flow, i.e., laminar or turbulent; usually 1 to 2 x 10 -3 ) 
c p = specific heat at constant pressure 
d = nozzle diameter 
F = thrust (lb) 

H a = specific enthalpy (total internal energy per unit mass) of ambient atmosphere 
He = specific enthalpy of unmixed jet 

Hj = specific enthalpy (total internal energy per unit mass) of mixed jet 
I sp = specific impulse (lb-sec lb -1 ). Varies approximately as T79K _1/2 . 

K p = chemical-equilibrium constant 
m = proportion of entrained atmosphere 
M = Mach number, unless otherwise noted 
9K = molecular weight 
P = pressure (lb in. -2 ) 

R = specific gas constant (ft-lb lb -1 °R -1 ), unless otherwise noted 


58 


AERIAL TARGETS 


59 


T — absolute static temperature in °R unless otherwise noted 
v = velocity (ft sec -1 ) 

Vm = the differential velocity between the initial jet and the atmosphere 
W = exhaust-gas weight flow rate (lb sec -1 ) 

y = r ^tio of specific heats (c p /c„); depends basically on the number of atoms in 
the molecule. For monatomic gases, y = 1.66; for diatomic gases, y ~ 1.30. 
p = air density outside boundary layer 
P = air velocity outside boundary layer 


Subscripts 

a = ambient 
c = combustion chamber 
e = nozzle exit plane 
t = nozzle throat 
w = wall 


F = 


Thrust and Mass Flow 

W c Wv e 

— £ =- £ + Ae(P e - P a ) = I sp W 

g g 


Exhaust Velocity 

v t = iygRT,) 2 

Ve ^ C = I sp g 

\_ 

M= v I {ygRT) 2 

<rc-r>] T 


Temperature 


T = Tc[l + :L y^M 2 ]"' 

= TAPIPc) iy - 1,1 y 


These three equations, relating temperature of the jet exhaust to internal condi¬ 
tions of the motor, are not precisely fulfilled in practice because of factors such as 
conductive and radiative heat losses, changing chemical composition, and nonideal- 
gas behavior.* Boynton and Neu [1] found that actual exhaust temperatures were 


*The energy budget of a typical rocket engine is as follows: 1% loss due to incomplete combustion, 
2% heat loss to engine walls, 27-57% in unavailable jet thermal energy, the balance in useful pro¬ 
pulsion energy and residual exhaust kinetic energy. 







60 


TARGETS 


best predicted by assuming shifting chemical equilibrium down to a certain pressure 
and frozen composition thereafter; this so-called ''freezing pressure” was determined 
by fitting to empirical data. Either process assumed alone would have incurred errors 
of about 200°K. 

Mixing and Afterburning. The energy balance of an exhaust jet of frozen compo¬ 
sition, mixing with the ambient atmosphere, has the following form [1]: 

_ H e + mH\ m(Vj — v m ) 2 
j 1 -fra 2g(l + m) 2 

More heat may be added to the jet by further reactions with the entrained atmosphere 
(afterburning). Table 4-1 gives the calculated heats released by afterburning processes 
for several fuels [2]. 


Table 4-1. Calculated Heat Release During Afterburning 


Propellant 

Combination 

O/F 
(by wt) 

Pc, Pc 
(psia)* 

%h 2 

(by wt) 

%CO 
(by wt) 

Heat Released 
(Btu/lb 
propellant) 

LO 2 /JP -4 

2.127 

600/14.7 

18.7 

37.7 

3117 

N 2 04/tUDMH, |-N 2 H 4 

2.01 

766/14 

3.1 

3.4 

358 

lo 2 /nh 3 

1.41 

600/14.7 

4.8 

— 

305 

lo 2 /c 2 h 5 oh 

1.5 

300/14.7 

10.2 

24.7 

1916 

lo 2 /n 2 h 4 

0.66 

600/14.7 

22.0 

— 

1540 

WFN A/aniline-alcohol 

3.0 

315/14.7 

5.1 

21.9 

1234 

RFNA/N 2 H 4 

1.159 

300/14.7 

14.4 

— 

914 

lo 2 /lh 2 

5.556 

600/14.7 

29.4 

— 

2830 

lf 2 /lh 2 

5.65 

600/14.7 

5.4 

— 

6420 

lf 2 /n 2 h 4 

2.4 

600/14.7 

0.3 

— 

20 

Solid C„H 2n - 1000/14.7 41.4 

Al, NH 4 C10 4 

* Under line indicates pressure to which the given gas composition pertains. 
tConsidering complete combustion of H 2 and CO. 

29.8 

3706 

Stagnation. Ideally, 

stagnation 

temperatures of an 

isentropic 

exhaust stream 


would equal the initial or chamber temperature, T c . Figure 4-1 illustrates repre¬ 
sentative stagnation temperatures [31. 

4.2.2. Jet Structure. The region behind a rocket or jet engine is an extremely 
complicated chemical and thermodynamic entity. This jet structure has been analyzed 
in great detail because knowledge of how much of what kind of gas is where and at what 
temperature makes possible calculations of the radiation field. This region, also called 
the flow field, is illustrated in Fig. 4-2. The flow-field characteristics are functions 
of both the engine and nozzle characteristics as well as the atmosphere in which the 
motor is operating. 














AERIAL TARGETS 


61 


Thrust in Vacuum 
Thrust at Sea Level 
Nozzle Area Ratio 
Nozzle Exit Diameter 
Propellants 
Chamber Pressure 
Nozzle Exit Pressure 
Nozzle Exit Static Temp 
Nozzle Exit Stagnation Temp 


171,000 lb 
150,000 lb 
8:1 

4.3.14 in. 

L0 2 -RP-1 
558 PSIA 
10.3 PSIA 
3400°F 

6000° F 250°F 



Fig. 4-1. Stagnation temperature (°F) distribution 
in exhaust at sea level. 


/ 



4.2.2.1. Flow Field Within the Atmosphere. Figure 4-2 actually illustrates the 
flow field for operation in the atmosphere for different nozzle expansion ratios and 
pictorially defines the shock, first period, and undisturbed cone. 

Aerodynamic heating is discussed in Chapter 21. 

Departure of the Jet from the Nozzle. 8 = the angle at the nozzle exit between 
the tangent to the nozzle and the tangent to the jet boundary. 


Underexpanded nozzle 

Pe > Pa 

8 > 0 

Optimally expanded nozzle 

P f + Pa 

8 > 0 

Overexpanded nozzle 

Pc < Pa 

8 < 0 


Shocks. For shock formation in supersonic jets, see Fig. 4-2. The theory of shock 
formation is covered in [5] and [6]. 

First Period. The first period of the jet is measured from the nozzle exit to the 
beginning of the reflected shock. See Table 4-2. 




































62 


TARGETS 


Table 4-2. Length in Nozzle Diameters [4] 


Exit Mach No. 



Pel P a 



CT 

li 

£ 

0.5 

1.0 

2.0 

4.0 

7.0 

1.5 

0 

1.2 

4.0 

5.0 

8.0 

2.5 

0.8 

2.4 

4.0 

5.6 

8.8 

3.5 

1.6 

3.4 

5.6 

8.4 

— 

4.5 

2.1 

4.4 

7.2 

— 

— 


Undisturbed Cone. The undisturbed cone is a region whose temperature, com¬ 
position, and length remain relatively constant for a given nozzle configuration (1 
= 3/2d) under all conditions of P a [7]. 

4.2.2.2. Flow Field in a Vacuum. Figure 4-3 illustrates the expansion of an ex¬ 
haust in a vacuum under several conditions [5]. 



Fig. 4-3. Exhaust expansion into a vacuum; line 
of constant Mach number [6]. 


4.2.3. Exhaust Composition 

4.2.3.I. Gases. Table 4-3 lists a number of liquid propellants and their combustion 
products. Lox/RP-1 exhaust composition vs mixture ratio is plotted in Fig. 4-4; Fig. 
4-5 is a similar plot for N 2 O 4 /0.5N 2 H4 + 0.5UDMH. 

Chemical Equilibrium. The balance of reactants and products achieved in chemi¬ 
cal equilibrium is expressed as 


K„ 


Pi • P 2 • P 3 ••• 
P, * P m * P„ •• 


where the P terms in the numerator are the partial pressures of the individual exhaust 
gases and the P terms in the denominator are the partial pressures of the reactant gases. 

Figure 4-6 [6] shows the equilibrium conditions as a function of temperature for 
a number of important reactions. 










MOLE PERCENT 


AERIAL TARGETS 


63 



Fig. 4-4. Exhaust composition 
of liquid oxygen/RP-1 vs mix¬ 
ture ratio. 


Fig. 4-5. Exhaust composition 
of N 2 C)4/0.5N 2 H< + 0.5UDMH 
vs mixture ratio. 



MIXTURE RATIO, O/F 







64 


TARGETS 


Table 4-3. Representative Liquid Propellants 


Oxidizer 

Fuel 



Conditions 



O/F 

Pc 

P; Ad A, 

T c 

T 

Isp 

Chlorine Trifluoride 

Hydrazine 

2.6 

300 

14.7;3.6 

6597°A 

4139°R 

258 

(CLF 3 ) 

(N 2 H 4 ) 



0.3;53 

6597°A 

1588 

330 

Fluorine 

Ammonia 

3.2 

300 

14.7;3.9 

7760 

5521 

312 

(F.) 

(NH 3 ) 



0.6;37 

7760 

2829 

403 

Fluorine 

Hydrazine 

2.2 

300 

14.7;3.9 

4415 

3179 

316 

(F 2 ) 

(N 2 H 4 ) 



0.6;39 

4415 

1710 

411 

Fluorine 

Hydrogen 

6.0 

100 

14.7;1.8 

5913 

4480 

303 

(F 2 ) 

(H 2 ) 



0.2;149 

5913 

815 

484 

98% Hydrogen Peroxide 

Hydrazine 

1.8 

300 

14.7;3.7 

5152 

3249 

253 

(H 2 0 2 ) 

(N 2 H 4 ) 



0.6;38 

5152 

1718 

325 

Nitrogen Tetroxide 

1/2 UDMH, 1/2 Hydrazine 

1.8 

800 

14.7;7.5 

5946 

3248 

282 

(N 2 0 4 ) 

[(CH 3 ) 2 N 2 H 2> N 2 H 4 ] 



0.8;66 

5946 

1830 

344 



2.0 

30 

14.7;1.02 

5389 

5079 

127 





0.3;76 

5389 

2143 

335 

Nitrogen Tetroxide 

Pentaborane 

3.2 

300 

14.7;4.1 

6812 

5298 

259 

(N 2 0 4 ) 

(B 5 H 9 ) 



0.3;98 

6812 

3754 

364 


Oxygen 

RP-1 

2.2 

800 

14.7;7.6 

6287 

3474 

291 

(0 2 ) 

(CioH 20 ) 



1.6;40 

6287 

2306 

339 

Oxygen 

Hydrazine 

0.8 

500 

14.7;5.4 

5915 

3567 

292 

(0 2 ) 

(N 2 H 4 ) 



1.0;38 

5915 

2068 

354 

Oxygen 

Hydrogen 

3.0 

300 

14.7;3.4 

4837 

2660 

364 

(0 2 ) 

(H 2 ) 



0.3;51 

4837 

1033 

462 



3.0 

50 

14.7;1.2 

4734 

3862 

251 





0.05;51 

4734 

1036 

461 

Oxygen 

Ethyl Alcohol 

1.5 

300 

14.7 

5705 

3170 

242 

(0 2 ) 

(C 2 H 5 OH) 







IRFNA 

UDMH 

2.8 

206 

14.7;2.9 

5400 

3857 

226 

(HN0 3 + N0 2 + H 2 0 + HF) 

((CH 3 ) 2 N 2 H 2 ) 



.21;54 


1075 

293 

Nitrogen Tetroxide 

15% Nitric Oxide, 85% 

2.1 

30 

14.7;1.02 

5454 

5135 

129 

(N 2 0 4 ) 

Mono Methyl Hydrazine 
[(NO, N 2 H 3 (CH 3 )] 



0.01;252 

5454 

1487 

359 


















AERIAL TARGETS 


65 


Table 4-3. Representative Liquid Propellants (Continued) 







Combustion Products (mole percent) 




h 2 o 

h 2 

H 

OH 

0 

NO 

n 2 

C0 2 

CO 

HF 

HC1 

Other 

— 

4.3 

0.2 

— 

— 

_ 

20.7 

_ 

_ 

56.1 

18.1 

0.61 Cl 

— 

4.1 

— 

— 

— 

— 

20.8 

- 

- 

56.3 

18.8 

- 

— 

1.33 

2.22 

_ 

_ 

_ 

14.3 

_ 

_ 

81.0 

_ 

1.11 F 

— 

1.19 

— 

— 

— 

- 

14.6 

— 

- 

83.5 

- 

- 

— 

1.82 

3.57 

_ 

_ 

_ 

20.1 

_ 

_ 

73.1 


1.4 F 

— 

2.97 

0.01 

- 

- 

- 

20.6 

- 

- 

76.4 

- 

- 

— 

50.4 

1.71 

_ 

_ 

_ 

_ 

_ 

_ 

47.9 

_ 

_ 

— 

51.7 

- 

- 

- 

- 

- 

- 

- 

48.3 

— 

— 

72.2 

6.48 

0.01 

0.02 

_ 

_ 

21.3 

_ 

_ 


_ 

_ 

72.2 

6.48 

- 

— 

— 

- 

21.3 

- 

- 

- 

- 

- 

42.3 

9.45 

0.01 

_ 

_ 

_ 

34.9 

7.15 

6.20 

_ 

_ 

_ 

38.8 

13.0 

— 

— 

— 

— 

34.9 

10.7 

2.70 

— 

— 

— 

38.3 

6.81 

2.30 

3.32 

0.84 

0.75 

33.5 

5.43 

6.93 

— 

— 

1.8 0 2 

44.5 

6.32 

- 

- 

— 

— 

36.0 

10.9 

2.19 

- 

- 

- 

3.07 

25.1 

6.62 

0.23 

0.04 

0.02 

22.3 

_ 

_ 

_ 

_ 

5.99 BO, 1.41 B 2 0 2 

1.01 

33.3 

1.15 

0.01 



23.7 





28.3 HB0 2 , 6.87 B 2 0 3 

0.47 BO, 0.56 B 2 0 2 

27.3 HB0 2 , 12.5 B 2 0 3 

32.1 

17.8 

0.04 

0.01 

_ 

_ 

_ 

14.3 

35.7 

_ 

_ 

_ 

26.7 

23.3 

— 

- 

- 

- 

— 

19.7 

30.3 

— 

- 

- 

53.4 

13.2 

0.05 

0.02 

_ 

_ 

33.3 

_ 

_ 

_ 

_ 

_ 

53.4 

13.2 

- 

- 

- 

— 

33.3 

- 

- - 

- 

- 

- 

37.8 

62.2 

__ 

_ 

_ 

_ 

_ 

_ 

_ 

_ 

_ 

_ 

37.8 

62.2 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

37.7 

61.9 

0.32 

0.01 

— 

— 

— 

— 

— 

— 

— 

— 

37.8 

62.2 

- 

— 

— 

— 

- 

— 

- 

- 

— 

— 

44.9 

10.2 

1.90 

3.37 

0.41 

- 

- 

13.8 

24.7 

- 

- 

0.67 0 2 

48.0 

6.24 

0.10 

0.08 

_ 

0.02 

22.0 

4.74 

17.5 

0.53 

_ 

_ 

35.3 

19.1 

0.59 

0.24 

0.01 

0.02 

22.0 

4.74 

17.5 

0.53 

- 

— 

34.8 

8.55 

2.88 

3.06 

0.76 

0.62 

32.7 

5.63 

9.82 

_ 

— 

1.18 0 2 

35.2 

13.7 

— 

— 

— 

— 

34.8 

14.7 

1.63 

— 

— 

— 




















66 


TARGETS 


C0 2 - CO + (l/2)0 2 



(a) 



(b) 

Fig. 4-6. Equilibrium constant K p vs temperature [6]. 






























































AERIAL TARGETS 


67 


Molecular Emission. Table 4-4 is a tabulation of the infrared emission bands of 
the principal exhaust gases. 

Table 4-4. Major Infrared Emission Bands of Exhaust Gases [8] 


Gas Approximate Center of Major Emission Bands (p) 

H 2 0 0.94, 1.14, 1.38, 1.88, 2.66, 2.74, 3.17, 6.27 

C0 2 1.96, 2.01, 2.06, 2.69, 2.77, 4.26, 4.68, 4.78, 4.82 

5.17, 15.0 

CO 4.663, 2.345, 1.573 

HC1 3.465, 1.764, 1.198 

NO 5.30, 2.672 

N0 2 4.50, 6.17, 15.4 

N 2 0 2.87, 3.90, 4.06, 4.54, 7.78, 8.57, 16.98 

OH 1.00, 1.03, 1.08, 1.14, 1.21, 1.29, 1.38, 1.43, 1.50, 

1.58, 1.67, 1.76, 1.87, 1.99, 2.15, 2.80, 2.94, 3.08, 3.25 
3.43, 3.63, 3.87, 4.14, 4.47 

S0 2 4.0, 4.34, 5.34, 7.35, 8.69 


4.2.3.2. Particles in Exhausts. Boynton [9] measured the carbon particle size dis¬ 
tribution in the exhaust of a small rocket engine. The distribution is shown in Fig. 4-7. 
It shows the apparent diameters in 22,000x micrographs. 



PERCENT OF PARTICLES SMALLER (Gaussian Scale) 


Fig. 4-7. Size distribution of carbon particles. 

Kurtovich and Pinson [101 measured the particle-size distribution in the exhaust 
stream of a scale-model solid-propellant rocket. In this case, the particles were oxide 
products of the pulverized aluminum used in some high-energy solid fuels. The 
distributions at two points downstream of the exit are given in Fig. 4-8. In Fig. 4-9 
the computed effect of particle size on the overall exhaust emissivity is given, for the 
distribution indicated by the solid line in Fig. 4-8. 

In Fig. 4-8 the particle size distribution at two distances from the nozzle exit plane 
is given. The volumetric particle concentration was 1.14 X 10 -5 . The solid curve 



FREQUENCY OF PARTICLES 


68 


TARGETS 



1 ---—a-*-L_^J-- 

0.1 0.2 0.5 1 2 5 10 20 50 100 


T 

c 

P 


c 


= 6000°R 
= 675 psia 


PARTICLE SIZE (microns) 



Fig. 4-8. Particle size distribution. 


Fig. 4-9. Emissivity as a function 
of upper limit of particle size. 


in Fig. 4-8 represents a distribution for which the emissivity of a particle cloud 1 ft thick 
has been calculated. This is shown in Fig. 4-9 as a function of an upper limit to particle 
size. 


4.2.4. Heating During Exit. Heating during exit, as a result of supersonic veloc¬ 
ities in the atmosphere, may be calculated, though certain data, particularly those 
regarding the boundary-layer flow, are not completely determined. The heat flow, 
q, into a surface is calculated as follows: 


q = ci,pfJ.c p 


T(l + R^-^M 2 


-T. w 


In this equation, T = boundary-layer temperature and R is the recovery factor (0.82 
to 0.88 [11] depending mostly on the state of flow). 

Radiation Processes. Aerodynamic heating is discussed in Chapter 21. Figure 
4-10 is an example of a time-temperature curve of a nose fairing. 


Nose Fairing 



31 Cal. Ogive 
0.045” Al. 



u 
0) 
c n 

> 

0) 

<v 

o 

£ 

>< 

H 

M 

u 

o 

j 

g 


Fig. 4-10. Temperature of Aerobee 150 
nose fairing (NASA Flight No. 4.12). 













































RADIOMETRIC ANALYSIS AND DISCUSSION 


69 

4.2.5. Heating During Reentry. Reentry heating is caused by aerodynamic 
friction and shock effects. The radiation contributions are thus from the heated body 
and the shocked air. 

As a body enters the earth’s atmosphere, considerable energy is released through the 
interaction of the body and the atmosphere. The optical radiations released are func¬ 
tions of the original condition of the event, e.g., velocity, mass, material, angle of attack, 
and vehicle shape. 

Below 70 or 80 km, the atmosphere may be considered to be a continuum, and fluid- 
dynamics principles may be applied. 

A blunt body has a comparatively large normal shock, with the result that consider¬ 
able air is heated and dissociated. Much of the energy stored during dissociation is 
not released until the air has passed into the wake. A slender vehicle, on the other 
hand, has little or no normal shock, so that there is less heating and little dissociation 
of the air. The frictional drag of the surface does cause the air in the boundary layer 
to decelerate, but the energy thus made available can in large part be efficiently con¬ 
ducted to the body. 

The radiation from the surface may be calculated if the temperature and spectral 
emissivity of the surface are known. Empirical relationships have been developed 
which permit one to calculate the heat input for spheres, cones, cylinders, and combina¬ 
tions thereof. These relationships depend upon vehicle velocity, ambient density, 
and of course the shape of the vehicle. The temperatures reached are functions of 
the thermal properties of the material of the body. However, the contributions from 
contaminants in the wake often completely overshadow all other contributions. 

4.3. Surface Targets 

Infrared surface targets include a wide variety of radiation sources which can be cat¬ 
egorized in a number of different ways. There are land targets and sea targets; there 
are strategic targets and tactical targets; there are fixed targets and mobile targets; 
there are active targets and passive targets. The wide differences between types of 
surface targets are evident from a listing of just a few examples: rivers, lakes, woods, 
fields, roads, railroads, airstrips, buildings, bridges, factories, shipping facilities, 
power plants, ships, submarine wakes, vehicles, and personnel. In spite of the diver¬ 
sity, it is apparent from this list that most surface targets are opaque, or nearly opaque, 
bodies and that many of them (though certainly not all) are at close to ambient tem¬ 
peratures (approximately 300°K). While some surface targets (e.g., the exhaust 
manifolds of tanks or other vehicles, power plants, or blast furnaces) can be distin¬ 
guished by temperatures which are definitely higher than any of the surroundings, 
many passive objects near ambient temperature must be recognized by other charac¬ 
teristics, such as shape, size, position, and contrast, in the spatial-distribution pat¬ 
tern of radiance. 

4.4. Radiometric Analysis and Discussion 

See Chapter 2 (Radiometry), Chapter 6 (Atmospheric Phenomena), and particularly 
Chapter 5 (Backgrounds) for more complete discussions of these topics. Note that 
atmospheric phenomena enter into the situation not only because of atmospheric 
attenuation of source radiation but also through the meteorological effects on the 
surface temperature and emissivity of radiation sources. This is strikingly illustrated 
by the "washout effect” [12,131. Chapter 5 is particularly pertinent because, in gen¬ 
eral, there is no inherent difference between a background source and a target source 
of radiation. The designation as one or the other reflects only the interest of the 
moment —the source of interest is a target; other sources, from which radiation is 


70 


TARGETS 


received along with target radiation, are the background. Also, for most applications 
the important considerations are those of target-background contrast, not just of tar¬ 
get radiation alone. 

The radiance N of an opaque body (which includes most surface targets) consists of 
that due to self-emission N e and that due to reflection or scattering of incident radi¬ 
ation N r : 

N = N e -f N r w cm -2 sr -1 (4-1) 

The relative magnitudes of N e and N r depend upon a number of factors, of which the 
two most important are: (1) the surface properties of the body (including the surface 
temperature) and (2) the incident radiation. N e will be more likely to predominate 
over N r when the surface has high emissivity and low reflectance (rough and dark), 
and conversely N r will tend to be greater when the surface has high reflectance and low 
emissivity (bright and shiny). The interrelationship is examined in more detail in 
Section 4.4.3. Considerations of specular vs diffuse reflectance and the relative spatial 
positions of illuminating source, target, and detector also may greatly affect the rela¬ 
tive magnitude of N r which, of course, depends directly upon the incident radiation, 
especially its magnitude and its distribution in wavelength and in direction. 

At night, self-emission N e usually predominates for military targets in the field 
and N r may be neglected, except when the target is irradiated by the source of an active 
or semiactive detection system (usually operating at wavelengths of 1 fx or less). In 
sunlight the relative importance of N e and N r depends greatly on the wavelength 
range as well as on target surface conditions (including temperatures). At longer 
wavelengths, greater than about 4 or 5 /x, reflected or scattered sunlight is relatively 
unimportant and N r may usually be neglected; at shorter wavelengths, less than about 
1 or 2 /x, self-emission becomes unimportant except for very hot targets, such as an 
exposed red-hot exhaust pipe, so that for bodies close to ambient temperatures N e 
may be ignored. At intermediate wavelengths, from about 1 to 5 /a, either or both 
N e and N r may be important. Reference [141 is an annotated bibliography on emit- 
tance and reflectance in the infrared, listing 910 pertinent references. 

4.4.1. Thermal Emission. The simplest, and probably the most frequently used, 
approach to the evaluation of N e is to consider an opaque solid as a graybody which, 
at a uniform surface temperature of T°K and in a direction in which its emissivity is 
e, will radiate according to the Stefan-Boltzmann law: 

N e = eaT 4 ln w cm -2 sr -1 (4-2) 

where cr = 5.67 X 10 -12 w cm -2 (°K)~ 4 . 

If the surface is rough or weathered, so that it approximates a perfectly diffusing or 
"Lambert law” surface, the emissivity e, and hence N e , will be the same for all directions, 
but in general it is a function at least of the angle 6 from the normal to the surface, e = 
e(0). If the surface is not uniform, it may also vary with azimuth direction <p and with 
position. The exact relationships are summarized in Section 4.4.3. Nevertheless, the 
Lambert law assumption is often made for lack of any data to establish the variations 
of € with respect to direction and/or position. 

If € is constant with respect to both direction and wavelength, and the emitting sur¬ 
face is at a uniform temperature T, the radiant intensity of the entire target in a given 
direction is 


Je = N e 


l 


cos 6 dA = N e A p 


w sr -1 


(4-3) 


RADIOMETRIC ANALYSIS AND DISCUSSION 


71 


where A p = / cos 0 dA is the projected area of the target perpendicular to the given 
direction (6 is the angle between that direction and the normal to the surface element 
dA and the integration is carried out over all of the "exposed” surface of the target, 
i.e. y all that can be "seen” from the given direction). If, however, there are temper¬ 
ature gradients, and different areas of the exposed surface are at different temperatures, 
the radiant intensity is given instead by 

J e = J N e cos 6 dA w sr _1 (4-4) 

where N e is expressed as a function of the temperature T of the element dA by Eq. (4.2) 
and the temperature T, in turn, is expressed as a function of the surface coordinates 
which define dA. The integration may be replaced by, or approximated by, a summa¬ 
tion if the surface can be divided into uniform regions each of which radiates in accord¬ 
ance with Eq. (4-2). In that case the radiant intensity of each exposed region is com¬ 
puted separately by Eq. (4-3) and the results are simply added to obtain the radiant 
intensity of the entire target. 

It is only when the target surface has constant emissivity and is also everywhere 
at uniform temperature that the value of its radiance N e , or its radiant intensity J e , 
as measured with a nonselective radiometer (with equal response at all wavelengths), 
can be unambiguously associated with its temperature. It is for this reason that the 
WGIRB (Working Group on Infrared Backgrounds) recommended strongly against the 
unfortunately all-too-common practice of using temperature as a radiometric unit 
in lieu of radiance [15]. 

Even when temperature gradients and differences exist and/or the emissivity is not 
constant with location and direction (but is constant with respect to wavelength), 
the radiance of each portion (possibly infinitesimal) of the surface of an opaque body 
is directly associated with the surface temperature of that portion by Eq. (4-2). It 
is then possible to evaluate or estimate the contrast between different areas with 
slightly different surface temperatures and/or emissivities by the relation 

dN e _ cr(4eT 3 dT 4- T 4 de)/n 
N P €crT 4 l7r 

— 4dTIT -I- dele dimensionless (4-5) 

Larger differences are evaluated by directly computing N e for each portion by means 
of Eq. (4-2) and then taking the difference or making the comparison. Contrast is 
then measured, by analogy with the expression dNIN, by 

C = OV, - N 2 )/N = 2(/V, - N 2 )/(N l + N 2 ) (4-6) 

If the background radiance N B is regarded as the reference level and one wishes to 
designate the contrast, with respect to this background level, of a target of radiance 
N t , one may instead compute the contrast as [16] 

C = (Nt — Nb)/N b dimensionless (4-7) 

It must be strongly reemphasized that the foregoing relations, in terms of total 
radiation (all wavelengths) have all been based on the graybody (constant-spectral- 
emissivity) assumption, which is usually only approximately true of real targets. 
Furthermore, application of these relations also implies a detector response to total 
radiation, i.e., nonselective response or constant spectral responsivity. Spectral 
distribution has been ignored, although the strong wavelength dependence of atmos- 
spheric attenuation and the frequent use of spectral filters and spectrally selective 




72 


TARGETS 


detectors makes it a very important consideration in many, if not most, applications. 
Accordingly, these relations will ordinarily give only approximate or qualitative re¬ 
sults for real situations, and often the approximation may be very poor. On the other 
hand, any evaluation of spectral effects as functions of source temperature, based on 
the Planck law (see Chapter 2), is best carried out in terms of each particular situ¬ 
ation. Attempts to write expressions which will apply generally in a wide variety 
of circumstances rapidly become prohibitively complex. 

4.4.2. Reflection. The measurement and specification of reflectance, even 
for opaque materials where multiple internal reflections are not involved, is not at all 
a simple matter. The additional complications which arise with semitransparent mate¬ 
rials are well discussed in [17] and [18], and will not be considered further here. 

It is particularly important here to recognize that the value of total reflectance, 
p, which is related by Kirchhoff’s law to the absorptance, a, or the emissivity, e, of 
an opaque body, 


a = e = (1 — p) dimensionless (4-8) 

may be quite different from the value of directional or partial reflectance (see Section 
2.10), which, together with the spatial distribution of incident radiation and the loca¬ 
tions of the target and detector, determines the value of N r in Eq. (4-1). The basic 
relationships involved are clearly and concisely summarized in an appendix to a paper 
by Richmond beginning on page 151 of [19]. 

In general, measured values of reflectance may be applied correctly only to situations 
which involve the same geometric relations between the source of irradiance, the 
reflecting surface, and the receiver or detector. Any change in the geometry of either 
the incident beam or the reflected beam of radiation may result in, or require, a dif¬ 
ferent value of reflectance. 

Possible confusion in terminology should be noted. Throughout most of [19], a dis¬ 
tinction is made between reflectivity, which is defined as the property of a material 
(i.e., measured with an ideally smooth and clean surface), and reflectance, which is the 
fraction of incident radiation reflected from a particular sample, regardless of its sur¬ 
face condition. The terms emissivity and emittance are used similarly with correspond¬ 
ing meanings, and the power per unit area emitted by a source, which we have called 
the radiant emittance, W (following [20]) is instead called the emissive power. Still 
another term, albedo, is frequently used by astronomers and meteorologists to desig¬ 
nate the total radiant reflectance of natural objects. Similarly, visual albedo is used 
to designate the luminous reflectance [21] (see Chapter 2 for further comments on 
this nomenclature). 

4.4.3. Factors Affecting the Temperatures of Passive Targets. Since the self¬ 
emission N e of an opaque target is so closely dependent upon its surface temperature, 
it is important to recognize the factors which determine that temperature for a passive 
or inert object under field conditions. This is a very complex situation that, in practice, 
is usually not amenable to quantitative treatment except for making very rough 
approximations [22]. However, qualitatively it is always useful and important to 
recognize and take into account the various contributing factors, such as the incident 
radiation or other source of heat, the recent history of incident radiation or other heat¬ 
ing, the absorptivity (see Eq. (4-8)), the heat conductivity, the heat capacity, the size 
and shape and material of the target, and other ambient conditions such as convective 
cooling or heating by winds or cooling by rain, or other precipitation or condensation 
(dew), and by its subsequent evaporation [23]. While the foregoing list may not be 


RADIOMETRIC ANALYSIS AND DISCUSSION 73 

exhaustive, it probably includes the most important parameters and is complete enough 
to suggest others which may be pertinent in a particular case. 

The forms in which heat energy reaches the surface of the earth, both from above 
and from below, are shown in diagrams of the daytime heat balance (Fig. 4-11), and the 
nighttime heat balance (Fig. 4-12) [24]. These are simplified diagrams of average 
or gross effects and do not take into account the local gradients which result from 
differences in heat capacity and/or heat conductivity, interactions between terrain 
configuration and sun (shadowing) or winds (sheltering), etc. The resulting varia¬ 
tions in surface temperatures give rise to most of the observed diurnal and seasonal 
variations and the effects of overcast histories. A thorough understanding of the 
phenomena involved is essential for the interpretation of strip maps obtained with 
infrared scanners and for the development of effective camouflage techniques. 


Solar Radiation 


Reflection 
from Clouds' 



Diffuse Scattering 


Radiation rj 
from Sky P 


Absorption 


Long-Wave Outgoing Radiation 


Solar Radiation 

Reflection 
from Ground 


Surface 

Supplied to the Ground 


Evaporation 


Convection 

Radiative Pseudo- 
Conduction 

Heat Conduction 


Fig. 4-11. Heat balance at the 
earth’s surface at midday ([24], 
taken from The Climate Near the 
Ground, by R. Geiger, Harvard 
University Press, 1950). 


Fig. 4-12. Heat balance at the 
earth’s surface at night ([24], 
taken from The Climate Near the 
Ground, by R. Geiger, Harvard 
University Press, 1950). 
























74 TARGETS 

4.5. Measured Values of Radiant Emissivity and Reflectance 

The complexities involved in the description and specification of the interrelated 
parameters of radiant emissivity and reflectance have been summarized in preceding 
sections, particularly in Chapter 2. The basic concepts and relations are not always 
clearly defined or applied in reports of measurement results, and a variety of measur¬ 
ing techniques is employed, with the result that uncertainties exist regarding many 
published values [25]. Some representative values of emissivities and/or reflectances 
are presented here, but it must be reemphasized that they can be regarded as accurate 
and strictly applicable only in situations where the ray geometry corresponds to that 
used in making the measurements, and also only where the surface conditions (weather¬ 
ing, corrosion, etc.) of the target or sample correspond to those of the sample measured. 
Usually, however, it is difficult if not impossible even to determine the degree to which 
these conditions are satisfied or reproduced. Often the pertinent information is not 
included in the measurement report, but, even when it is, verification is not a simple 
matter. Consequently, the reported values are often useful only to indicate the orders 
of magnitude involved and to suggest the probable relationship or contrast between 
a target and other targets or background objects as detected by a particular infrared 
device. 

It is only possible to present here a few representative curves and tabulated values. 
Additional measurement reports are found in the literature, and an extensive listing 
of published material prior to 1959 is found in [14]. Reference [26] contains an ex¬ 
tensive tabulation of published experimental results through 1957, giving considerable 
detail about the coverage of the measurements and references to the measurement 
reports but not the measured values themselves. More detailed data on the reflectance 
and emissivity of different materials, particularly of pure substances with clean sur¬ 
faces, measured in the laboratory (reflectivities and emissivities rather than reflec¬ 
tances and emittances or emissances, in the much-debated terminology for distinguish¬ 
ing material properties from the parameters of particular samples [19]) are found in 
standard references, such as the Handbook of Chemistry & Physics (Chemical Rubber 
Publishing Co.) and the International Critical Tables. 

The measurements shown here are all for opaque substances, and the reflectances 
are assumed to be at least approximately equivalent to total reflectance p, or directional 
reflectance pa, as defined in Chapter 2 (note particularly the distinction between the 
latter and the partial reflectance or reflectance distribution function p') so that they 
may be related to the corresponding emissivities and absorptances by Eq. (4-8). As 
emphasized in Section 4.4.3., however, Eq. (4-8) holds strictly only for monochromatic 
radiation, for radiation consisting only of wavelengths for which values of reflectance, 
emissivity, and absorptance do not vary with wavelength, or where the reflectance 
and absorptance values are those for incident radiation with a blackbody (graybody) 
spectral distribution. 

Spectral reflectance curves for a few varieties of ordnance materials are presented 
in Figs. 4-13 to 4-16 [27]. Similar curves for surfaces and finishes of naval interest 
are shown in Figs. 4-17 through 4-24 [28]. Reference [29] covers an extensive study 
of the reflectances of a wide variety of terrain features and of paints and finishes, in¬ 
cluding the effects of water immersion on the latter. Some of the spectral reflectance 
curves are presented in Fig. 4-25 as they were summarized in [30], where these and 
other similar data from [29] were used to compute the emissivities in the 3-5-p. and 
8-13-/Lt bands listed in Table 4-5, which also includes values, obtained similarly from 
the spectral reflectance curves of [28], for reflectances in the 0.7-1.0-/Z band and emis¬ 
sivities in the 1.8-2.7-p. band. The importance of the substrate to which a paint is 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 75 


applied and the effect that the substrate may have on the spectral reflectance of the 
painted surface is illustrated by Figs. 4-26 and 4-27 [29]. 

Curves of directional reflectance for some paving and roofing materials and paints, 
showing the effects of different angles of incidence in different spectral regions, appear 
in Figs. 4-28 through 4-33 [17]. Note that the "directional reflectivity,” r a p, of [17] 
(plotted in these figures) is related to the partial reflectance p', as defined in Eq. (2-45), 
as follows: 


r al 3 = p'(a, <f>, /3, (f> ± 7 t) cos a cos (i (4-9) 


Table 4-5. Reflectance (p) and Emissivity (e) of Common Terrain Features* 



0.7-1.Op 

1.8-2.7 p 

CO 

1 

cn 

8-13 p 

Green Mountain Laurel 

p = 0.44 

€ = 0.84 

€ = 0.90 

€ = 0.92 

Young Willow Leaf (dry, top) 

0.46 

0.82 

0.94 

0.96 

Holly Leaf (dry, top) 

0.44 

0.72 

0.90 

0.90 

Holly Leaf (dry, bottom) 

0.42 

0.64 

0.86 

0.94 

Pressed Dormant Maple Leaf (dry, top) 

0.53 

0.58 

0.87 

0.92 

Green Leaf Winter Color — Oak Leaf (dry, top) 

0.43 

0.67 

0.90 

0.92 

Green Coniferous Twigs (Jack Pine) 

0.30 

0.86 

0.96 

0.97 

Grass —Meadow Fescue (dry) 

0.41 

0.62 

0.82 

0.88 

Sand—Hainamanu Silt Loam — Hawaii 

0.15 

0.82 

0.84 

0.94 

Sand —Barnes Fine Silt Loam —So. Dakota 

0.21 

0.58 

0.78 

0.93 

Sand —Gooah Fine Silt Loam —Oregon 

0.39 

0.54 

0.80 

0.98 

Sand — V ereiniging — Africa 

0.43 

0.56 

0.82 

0.94 

Sand —Maury Silt Loam — Tennessee 

0.43 

0.56 

0.74 

0.95 

Sand —Dublin Clay Loam —California 

0.42 

0.54 

0.88 

0.97 

Sand —Pullman Loam —New Mexico 

0.37 

0.62 

0.78 

0.93 

Sand —Grady Silt Loam —Georgia 

0.11 

0.58 

0.85 

0.94 

Sand —Colts Neck Loam —New Jersey 

0.28 

0.67 

0.90 

0.94 

Sand — Mesita Negra — lower test site 

0.38 

0.70 

0.75 

0.92 

Bark —Northern Red Oak 

0.23 

0.78 

0.90 

0.96 

Bark —Northern American Jack Pine 

0.18 

0.69 

0.88 

0.97 

Bark —Colorado Spruce 

0.22 

0.75 

0.87 

0.94 


*Estimated average values of reflectance p, or emissivity e = 1 — p, in the indicated wavelength bands, read from 
the spectral reflectance curves of [29] (some of which are shown in Fig. 4-25). 


Fig. 4-13. Spectral reflectance of 
rubber track block, natural rubber 
on nylon fabric, CF-11 [27]. 







76 


TARGETS 



Fig. 4-14. Spectral reflectance of 
cotton herringbone uniform fabric, 
D.C. No. 7 Wool Elastique, O.D. 
No. 51 [21]. 


Fig. 4-15. Spectral reflectance of 
mild steel oxidized blue and stain¬ 
less steel oxidized grey [27], 




Fig. 4-16. Spectral reflectance of 
butyl on cotton fabric, O.D., CF-21 
and vinyl on cotton fabric, O.D., 
CF-13 [27], 









REFLECTIVITY (%) REFLECTIVITY (0) 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 77 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH(m) 





































































7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH (jz) 


Fig. 4-17. Spectral reflectance of aluminum foil, 0.001 in. RM-216 
(Reynolds Metal Co.) [28]. 








































REFLECTIVITY (%) REFLECTIVITY (%) 


78 


TARGETS 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH ( 4 ) 


100 


50 


0 

7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH ( 4 ) 



Fig. 4-18. (a) Spectral reflectance of aluminum, asphalt base. No. 3483 

(Sears Roebuck and Co.), and (6) spectral reflectance of aluminum lacquer, 
No. S-2432-C (Stoner-Mudge Inc.) [28]. 















































REFLECTIVITY (%) REFLECTIVITY (%) 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 79 



WAVELENGTH ( 4 ) 



WAVELENGTH ( 4 ) 


Fig. 4-19. (a) Spectral reflectance of Kerpo No. 25 aluminum, and (6) 

Kerpo spray coat, gold (Protective Coatings Corp.) [28]. 









































REFLECTIVITY (%) REFLECTIVITY (%) 


TARGETS 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH (/jl) 



WAVELENGTH (/i) 


Fig. 4-20. Spectral reflectance of enamel, Chinese red, interior, 
No. 2816 decoret enamel (W. P. Fuller and Co.) [28]. 







































REFLECTIVITY (%) REFLECTIVITY (%) 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 81 




7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH (m) 

Fig. 4-21. Spectral reflectance of enamel, white, exterior No. 175 
(Walter N. Boysen Co.) [28]. 











































REFLECTIVITY (%) REFLECTIVITY ( %) 


82 


TARGETS 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH (/i) 



7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH (a) 

Fig. 4-22. Spectral reflectance of (a) ideal masonry, No. 150 (red), (6) ideal 
masonry, No. 170 (green), exterior masonry paint (Ideal Chemical Products Inc.) 
[28]. 




































REFLECTIVITY (%) REFLECTIVITY (%) 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 83 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH (n) 

























































— 1- - _l 1 —- 






7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH (p) 

Fig. 4-23. Spectral reflectance of asphaltic road material, SC-4 
(Standard Oil Co. of California) [28]. 







































REFLECTIVITY (%) REFLECTIVITY (%) 


84 


TARGETS 



1.8 2.0 3.0 4.0 5.0 6.0 7.0 7.4 

WAVELENGTH (m) 



7.4 8.0 9.0 10.0 11.0 12.0 13.0 

WAVELENGTH (n) 

Fig. 4-24. Spectral reflectance of coal tar pitch, melting point 170-180°F 
(Barrett Div., Allied Chemical and Dye Corp.) [28]. 










































SPECTRAL REFLECTANCE 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 85 







Fig. 4-25. Spectral reflectance of six terrain features in the 1-15-p. region 
(based on [30]; original source) [29]. 





















On Black Paint No. 13 ° Perkin-Elmer 

Colored Zinc Oxide I D Beckman 

Double Pass Instrument 
(Sulfur Standard) 


86 


TARGETS 


i 


o 

to 


o 

T 



X 

H 

O 

Z 

u 

J 

w 

> 

< 

£ 



o 

o 


I I I I L 

o o o 

CO CO rr 


1 


a. 

X 

H 

o 

Z 

w 

w 

> 

< 

£ 


CO 


D 

o 
o 


L 

o 

5 


o 

CO 


(%) NOUD31J3H 


(%) NOIX03133H 


O 

CM 


Fig. 4-26. Spectral reflectance of paint No. 13, showing dependence on substrate [29]. 










On Black Paint No. 2 o Perkin-Elmer 

Red Synthetic Iron Oxide (Spheroidal) II a Beckman 

Double Pass Instrument 


MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 


87 




(%) NOI103333H 


(%) NOI1D3333H 


Fig. 4-27. Spectral reflectance of paint No. 2, showing dependence on substrate [29]. 















88 


TARGETS 





Fig. 4-28. Directional reflectivity (Eq. 4-9) 
curves for concrete [17]. 



































MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 89 



Thermistor 







































90 


TARGETS 



Fig. 4-30. Directional reflectivity (Eq. 4-9) curves 
for concrete painted with Codit silver paint [17]. 



































MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 91 


o° 




SOURCE ANGLES 

■ 0 * 

A 20* Open Symbol - Theoretical 
• 40* Closed Symbol - Experimental 

*60* 


Fig. 4-31. Directional reflectivity (Eq. 4-9) curves for concrete 
painted with Centerlite white paint (abraded) [17]. 




























92 


TARGETS 



C/3 

w 

J 

o 


u 

u 

OS 

D 

o 

co 


cU 

c 

a ® 

3 g 

£ s 

8 8 * 

.c w 

Eh ■ 

i H 

o 5 

1E 
6 >> 
>,co 
W-o 

c 4> 

8.1 

oo 


,666 

O OJ Tf ® 

« ^ * 




Fig. 4-32. Directional reflectivity (Eq. 4-9) curves for 
concrete painted with Flex-o-lite beaded paint [17]. 






































MEASURED VALUES OF RADIANT EMISSIVITY AND REFLECTANCE 93 


o' 



o' 



SOURCE ANGLES 

■ 0 * 

▲ 20* Open Symbol - Theoretical 
• 40* Closed Symbol - Experimental 

*60* 


Fig. 4-33. Directional reflectivity (Eq. 4-9) curves 
for corrugated metal [17]. 

























94 TARGETS 

References 

1. F. P. Boynton and J. T. Neu, Rocket Plume Radiance, V. Calculation of Adiabatic Flame Tem¬ 
peratures of Afterburning Rocket Exhaust, Rept. ERR-AN-011, Convair (Astronautics Divi¬ 
sion), San Diego, Calif. (May 1960). 

2. W. H. Krase, Exhaust-Gas Composition and Afterburning Energy Release for Selected Rocket 
Propellants, Memo RM-3100-PR, Rand Corp. (April 1962). 

3. Design Information Manual No. 101, Vol. I, Report SA57-53, Aerojet-General Corp. (Nov. 

I, 1957). 

4. Rocket Exhaust Patterns, Rept. No. 1844, Aerojet-General Corp. (Aug. 1960). 

5. Determination of the Envelope and Lines of Constant Mach Number for an Axially Symmetric 
Free Jet, Rept. ZJ-7-054, Convair Division, General Dynamics (March 14, 1958). 

6. G. P. Sutton, Rocket Propulsion Elements, Wiley, New York (1956). 

7. Seth L. Tuttle, "A Model for the Radiation Field of the Atlas Sustainer at High Altitudes,” 
AMRAC Proceedings, Vol. V, Part I, p. 69 (Nov. 1961) (SECRET). 

8. G. Herzberg, Infrared and Raman Spectra of Poly atonic Molecules, Van Nostrand, New York 
(1945). 

9. F. P. Boynton, Rocket Plume Radiance, IV. Studies of Carbon Particles Formed by Small 
Hydrocarbon Fueled Rocket Engines, Rept. ERR-AN-007, Convair (Astronautics Division), 
San Diego, Calif. (April 7, 1960). 

10. D. D. Kurtovich and G. T. Pinson, How to Find the Exhaust Heat Radiation of Aluminized 
Solid Rockets, Space Aeronautics (July 1961). 

11. Transactions of B AMIR AC 1960 Summer Study, Rept. No. 3768-6-X, Willow Run Laboratories, 
The University of Michigan, Ann Arbor, Mich. (Oct. 1960) (SECRET). 

12. W. R. Fredrickson, N. Ginsburg, and R. Paulson, Infrared Spectral Emissivity of Terrain, 
WADC-TR-58-229, Syracuse University (April 30, 1958) AD 155 552. 

13. N. Ginsburg, W. R. Fredrickson, and R. Paulson, "Measurements with a Spectral Radiometer,” 

J. Opt. Soc. Am., 50, 1176 (Dec. 1960). 

14. Dorothy E. Crowley, Emittance and Reflectance in the Infrared; An Annotated Bibliography, 
Rept. 2389-15-S, Willow Run Laboratories, The University of Michigan, Ann Arbor, Mich. 
(April 1959). 

15. Infrared Target and Background Radiometric Measurements — Concepts, Units, and Techniques 
(Report of WGIRB), Rept. 2389-64-T, Institute of Science and Technology, The University of 
Michigan, Ann Arbor, Michigan (April 1959). 

16. W. E. K. Middleton, "Vision Through the Atmosphere,” in Handbook der Physik, ed. by Flugge, 
Vol. XLVIII, pp. 254 and 262, (Springer-Verlag (1957)). 

17. A. LaRocca, J. Livisay, C. Miller, and G. Zissis, Characteristics of a Ground Infrared Range, 
Rept. No. 2849-6-F, Willow Run Laboratories, The University of Michigan, Ann Arbor, Mich. 
(June 1959) (SECRET) AD 314 492. 

18. H. O. McMahon, "Thermal Radiation from Partially Transparent Reflecting Bodies,” J. Opt. 
Soc. Am., 40, 376 (J\ine 1950). 

19. Henry H. Blau and Heinz Fischer, Radiative Transfer from Solid Materials, Macmillan, New 
York (1962). 

20. Committee on Colorimetry, Optical Society of America, The Science of Color, Crowell, New 
York, pp. 176-178 (1954). 

21. Air Force Cambridge Research Center, Handbook of Geophysics for Air Force Designers, 
Chap. 2 and 14. 

22. G. T. Pullan, "Temperature and Emissivity Measurements,” Infrared Characteristics of RCN 
Destroyer-Escorts, CARDE TR-413/62 (July 1962). 

23. Geophysics Research Directorate, Air Force Cambridge Research Center, U. S. Air Force Hand¬ 
book of Geophysics, revised ed., Macmillan, New York, pp. 215-220 (1960). 

24. C. E. Heerema and H. C. Graboske, "Interpretation of Infrared Images,” Proc. IRIS, IV, 4, 
393-403 (Oct. 1959). 

25. R. K. McDonald, "Techniques for Measuring Emissivities,” Proc. IRIS, V, 3, 153 (July 1960). 

26. H. H. Blau, Jr., J. L. Miles, and L. E. Ashman, Thermal Radiation Characteristics of Solid Mate¬ 
rials; A Review, Scientific Report 1, AFCRC-TN-58-132, Arthur D. Little, Inc. (March 31,1958). 
AD 146 883. 

27. David K. Wilburn and Otto Renius, The Spectral Reflectance of Ordnance Materials at Wave¬ 
lengths ofl to 12 Microns, The Detroit Arsenal, Final Rept. No. 3196 (Feb. 1955). AD 087 246. 

? p W. L. Starr, E. R. Streed, and A. I. Funai, Principles of Infrared Camouflage for Low Tempera¬ 
ture Targets, Tech. Note N148, U. S. Naval Civil Engineering Research and Evaluation Labor¬ 
atory, Port Hueneme, Calif. (July 21, 1953). 

29. Max Kronstein, Research Studies and Investigations on Spectral Reflectance and Absorption 
Characteristics of Camouflage Paint Materials and Natural Objects, Final Report, New York 
University (March 1955). AD 100 058. 

30. G. D. Currie et al.. Infrared Aerial Reconnaissance Interpretation, Bendix Corp., Systems 
Division, Arm Arbor, Mich., Note BSR-175, RADC-TN-60-29 (January 24, 1960) (CONFI¬ 
DENTIAL). AD 315 971. 


Chapter 5 
BACKGROUNDS 


Richard Kauth 

The University of Michigan 


CONTENTS 


5.1. Sky Backgrounds. 96 

5.2. Aurora. 100 

5.2.1. Auroral Spectra. 100 

5.2.2. Auroral Zones. 101 

5.2.3. Periodic Variations. 102 

5.2.4. Height and Vertical Extent. 102 

5.3. Night Airglow. 104 

5.4. Stellar Radiation. 107 

5.4.1. Stellar Magnitudes. 107 

5.4.2. Stellar Spectral Classes. 107 

5.4.3. Number of Stars. 107 

5.4.4. Galactic Concentration of Stars. 109 

5.4.5. Spectral Distribution of Stellar Radiation. 110 

5.4.6. Determining Spectral Irradiance of Celestial Bodies. 112 

5.5. The Earth as a Background. 115 

5.5.1. Geometric Relationships. 115 

5.5.2. Path Lengths. 116 

5.6. Cloud Meteorology. 118 

5.6.1. Cirrus Clouds. 122 

5.6.2. Stratospheric Clouds. 124 

5.6.3. Probability of Coverage at Various Altitudes. 124 

5.7. Stratospheric Aerosols. 141 

5.8. Spectral Radiance of Terrain. 142 

5.8.1. Terrain Temperature. 142 

5.8.2. Terrain Emissivity and Reflectivity. 142 

5.8.3. Spectra in the Emission Region. 143 

5.8.4. Spectra in the Scattering Region. 145 

5.8.5. Spectral Radiance of Various Objects and Surfaces. 145 

5.9. Marine Backgrounds. 166 

5.9.1. Infrared Optical Properties of Sea Water. 166 

5.9.2. Sea-Surface Geometry. 168 

5.9.3. Sea-Surface Temperature Distribution. 169 

5.9.4. Sky Radiance. 170 

Notes Added in Proof. 170 


95 






































5. Backgrounds 


5.1. Sky Backgrounds* 

Sky-background radiation in the infrared is caused by scattering of the sun’s radiation 
and by emission from atmospheric constituents. Figure 5-1 illustrates the separation 
of the spectrum into two regions —the solar scattering region short of 3 /x, and the 
thermal emission region beyond 4 p,. Solar scattering is represented by reflection from 
a bright sunlit cloud, and alternatively by a curve for clear-air scattering. The thermal 
region is represented by a 300°K blackbody. Figure 5-2 shows blackbody curves for 
temperatures ranging from 0° to 40°C. This simple model is modified by a number of 
factors: in the solar region there are absorption bands of water vapor at 0.94, 1.1, 1.4, 
1.9, and 2.7 fx, and of carbon dioxide at 2.7 fx. The effect of these bands is shown in 
Fig. 5-3. 

In the thermal region the bands which have strong absorption, and thus strong emis¬ 
sion, will approach very closely to the blackbody curve appropriate to the temperature 
of the atmosphere. Less strongly emitting regions may contribute only a small frac¬ 
tion of the radiation of a blackbody at the temperature of the atmosphere. The bottom 
curve in Fig. 5-4 is a good example. This zenith measurement, taken from a high, 
dry location, Elk Park, Colorado, shows low emission except in the strong band of 
CO 2 at 15 fx and of H 2 0 at 6.3 /x. There is also a weak emission peak, due to ozone, 



WAVELENGTH (ji) 

Fig. 5-1. Contributions from scattering 
and atmospheric emission to background 
radiation [1]. 



WAVELENGTH (p) 

Fig. 5-2. Spectral radiance of a 
blackbody with temperature in the 
range of 0 to 40°C [1]. 


*See also Notes Added in Proof, page 170. 


96 








SKY BACKGROUNDS 


97 



WAVELENGTH (ju) 

Fig. 5-3. Spectral radiance of 
the clear daytime sky [1], 


^ 1500 

tH 

I 

Sh 

C0 

rq 

i 

s 

° 1000 
£ 
a. 



WAVELENGTH (p) 

Fig. 5-4. The spectral radiance of 
a clear nighttime sky for several 
angles of elevation above the horizon 
(Elk Park Station, Colorado) [1], 


at 9.6 fi. The low-level continuum is due to the wings of the strong bands of H 2 0 
and C0 2 . The effect of increased humidity and air mass can be seen by comparing 
the bottom curves of Fig. 5-4 and 5-5. Figure 5-5 shows measurements taken at a 
humid sea-level location, Cocoa Beach, Florida. 

The effect of increasing air mass alone can be seen in both Fig. 5-4 and 5-5, by com¬ 
paring curves taken from the same altitude at various elevation angles. The emission 
shows a systematic decrease with increasing 
elevation angle. The direction of look also has 
an effect in the solar scattering region, as seen 
in Fig. 5-3, where, for a clear sky, the sun posi- jt 

tion is fixed and the spectral radiance is plotted 
for several observer angles. 

The position of the sun has a strong effect on 
the scattered radiation in the solar region, as 
shown in Fig. 5-6, where the observer looks at 
the zenith and the elevation angle of the sun 
is varied but has little effect on the radiation 
in the thermal region. The temperature of 
the atmosphere, on the other hand, has a strong 
effect on the radiation in the thermal region 
but little effect in the solar region. The pres¬ 
ence of clouds will affect both the near-infrared 
solar scattering and the thermal-region 
emission. 

Near-infrared radiation exhibits strong 
forward scattering in clouds. Thus the rela¬ 
tive positions of sun, observer, and cloud 
become especially important. For a heavy 
overcast sky, multiple scattering reduces the 
strong forward scattering effect. 



Fig. 5-5. The spectral radiance of 
a clear nighttime sky for several 
angles of elevation above the horizon 
(Cocoa Beach, Florida) [1], 
























98 


BACKGROUNDS 



WAVELENGTH (pi) 


Fig. 5-6. Spectral radiance of a 
clear zenith sky as a function of 
sun position; A = sun elevation 77°, 
temperature 30° C; B = sun elevation 
41°, temperature 25.5°C; C = sun 
elevation 15°, temperature 26.5°C 
[ 1 ]. 


- Cloud Spectrum 

- Blackbody Curves 



Fig. 5-7. The spectral radiance of the 
underside of a dark cumulus cloud [1]. 


Thick clouds are good blackbodies. Emission from clouds is in the 8-13-/X region 
and is, of course, dependent on the cloud temperature. Because of the emission and 
absorption bands of the atmosphere at 6.3 /x and 15 n, a cloud may not be visible in these 
regions and the radiation here is determined by the temperature of the atmosphere. 
A striking example is given in Fig. 5-7. Here the atmospheric temperature is +10°C 
and the radiation in the emission bands at 6.3 /x and 15 /x approaches a value appro¬ 
priate to that temperature. The underside of the cloud has a temperature of — 10°C, 
and the radiation in the 8-13-/X window approaches that of a blackbody at — 10°C. 

Figure 5-8 shows the variation of sky radiance as a function of elevation angle. 
Figure 5-9 shows the variation with respect to variations of ambient air temperature, 
and Fig. 5-10 shows seasonal variations. 










SKY BACKGROUNDS 




Fig. 5-8. The spectral radiance of 
sky covered with cirrus clouds at 
several angles of elevation [1]. 


Fig. 5-9. Zenith sky spectral radi¬ 
ance showing the large variation 
with ambient air temperature [1]. 



Fig. 5-10. Spectral radiance of overcast skies in winter 
and summer [2]. 












100 

5.2. Aurora [3,41- 


BACKGROUNDS 


5.2.1. Auroral Spectra.* Aurora emission lines occur at 0.92, 1.04, and 1.11 n; 
the measured brightnesses are about 6 X 10 -8 w cm -2 sr -1 line -1 [5]. 

Figure 5-11 shows the auroral spectrum between 0.9 and 1.2 /x. This reproduction 
was obtained by averaging a number of individual spectra [5]. Figure 5-12 shows 
auroral spectra between 1.4 and 1.65 /x. The dotted curve is the airglow spectrum 
fitted to the auroral spectrum in a region where the auroral emission appears feeble. 
Spectra (a), (6), and (c) were made in consecutive scans, with a total time of 3 min. 
The relative intensities of features on a single scan are not significant, since the aurora 
fluctuates in brightness during the scanning period [6]. 



Fig. 5-11. Auroral spectrum, 0.9 to 1.2 fi, 
obtained with a lead sulfide spectrometer; 
projected slit width, 100 A [5]. 


Fig. 5-12. Auroral spectra, 1.4 to 
1.65 pt, obtained with a lead sulfide 
spectrometer; projected slit width 
200 A [6], 




*See also Notes Added in Proof, page 170. 























AURORA 


101 


5.2.2. Auroral Zones [4]. Figure 5-13 shows the auroral zones. These are di¬ 
vided into 3 areas: the north and south auroral regions extending from geomagnetic 
latitudes 60° to the poles, the subauroral belts between 45° and 60°, and the minauroral 
belt between 45° N and 45° S. 

The auroral regions include the auroral zones, which are the regions of maximum 
occurrence, and the auroral caps, which are the polar regions within the auroral zones. 

Although aurorae occur primarily in the auroral regions, large displays may occur 
in quite low latitudes. However, in tropical and even low temperate latitudes they 
Eire extremely rare. 



Fig. 5-13. The hemisphere centered on (a) north¬ 
ern geomagnetic pole (78°5'N, 69°W geographic) 
and (6) southern geomagnetic pole (78°5'S, 
111°E geographic) [4]. 































102 


BACKGROUNDS 


The frequency of auroral occurrences has a maximum some 20° or 25° from the geo¬ 
magnetic poles. Figure 5-14 shows the geographic distribution of the frequency of 
aurorae in the northern hemisphere [7]. The isochasms refer to the number of nights 
during the year in which an aurora might be seen at some time during the night, and 
in any part of the sky, if clouds and other factors affecting visual detection of aurorae 
do not interfere. Figure 5-15 shows the zone of maximum auroral frequency in the 
southern hemisphere [8]. 



Fig. 5-14. Geographic distribution of the frequency of aurorae in the northern hemisphere [7]. 


5.2.3. Periodic Variations. The number of aurorae observed from a particular 
point over the course of a year may vary widely and is strongly correlated with solar 
activity. Minimum auroral activity corresponds with minimum solar activity. Max¬ 
imum auroral activity usually occurs about two years after sunspot maximum. 

5.2.4. Height and Vertical Extent. On auroral arcs and bands the most convenient 
height to measure is the apparent lower border, which is fairly sharp. An example 
of a set of such measurements in and near the auroral zone is shown in Fig. 5-16 [9]. 

















AURORA 


103 


The total number of measurements shows a concentration between 95 and 110 km, with 
a double peak. The lower limits of individual rays appear 10 or 15 km higher than the 
lower edges of most arcs, bands, and draperies. Sunlit auroral rays appear systemati¬ 
cally higher than displays in the dark atmosphere. Figure 5-17 shows the heights 
of rays over southern Norway. A few sunlit rays extend higher than 1000 km. 



Fig. 5-15. Zone of maximum auroral frequency in the southern hemisphere [8]. 



20 20 40 60 20 20 40 20 40 60 80 100 


POINTS MEASURED 

Fig. 5-16. Distribution of heights of lower borders 
of auroral arcs [9]. 


























104 


BACKGROUNDS 




Fig. 5-17. Length and position in the atmosphere of the vertical projections 
of auroral rays in (a) sunlight and (6) earth’s shadow (1917-1943) [10]. 


5.3. Night Airglow 

Airglow may be defined as the nonthermal radiation emitted by the earth’s atmos¬ 
phere, with the exceptions of auroral emission and radiation of a cataclysmic origin, 
such as lightning and meteor trails [4]. 

Night airglow emissions in the infrared are caused by transitions between vibrational 
states of the OH radical. The exact mechanism of excitation is still unclear, but 
the effect is to release energy from solar radiation stored during the daytime. Air¬ 
glow occurs at all latitudes. 

There is evidence [11] that some of the excitation is 

H + 0 3 -*> OH + 0 2 


OH + O —► 0 2 + H 






































































































































NIGHT AIRGLOW 


105 


Thus, it appears that the distribution of night airglow is related to that of ozone. 
The measured heights of the airglow range from 70 to 90 km, which corresponds to the 
location of ozone. 

Airglow brightness is specified in rayleighs (R) a measure of the apparent number 
of photons emitted in a column 1 cm 2 in diameter along the observer’s line of sight. 

1 R = 10 6 (apparent) photon/cm 2 -sec (column) = 47 tI 
where I is millions of photon/(cm 2 -sec-sr). 

To a good approximation, the nightglow increases away from the zenith as sec 0. 
Measurements usually are reported normalized to the zenith. 

Variations in airglow intensity during the night seem to be caused by the motion of 
large patches (airglow "cells”) with dimensions of about 2500 km moving with velocities 
of about 70 m sec -1 [12]. 

Figure 5-18 shows the relative brightness of airglow intensity [13]. Airglow emis¬ 
sions due to OH appear as small maxima in the vicinity of 1.6 and 2.15 fi. Although 
further emission bands are predicted in the range from 2.8 to 4.5 /x, they are thoroughly 
masked by the thermal emission of the atmosphere. Figure 5-19 shows the nightglow 
spectrum in the l-2-/x region [14]. Looking straight down from a satellite, the atmos¬ 
pheric spectrum should be very similar to that shown in Fig. 5-18 and 5-19. Table 
5-1 compares the approximate rates of emission for various airglow and auroral lines. 
The references in the footnotes should be consulted for further details. Note that, for 
the airglow, all results are given for the zenith itself rather than for the angles at which 
observations are usually made. 




WAVELENGTH (jut) 


Fig. 5-19. Nightglow spectrum, ob¬ 
tained with a scanning spectrometer 
(projected slit width 200 A). The ori¬ 
gins and expected intensities of OH 
bands are shown by vertical lines; the 
horizontal strokes indicate the reduc¬ 
tion due to water vapor [14]. 



















106 


BACKGROUNDS 


Table 5-1. Comparison of Aurora and Airglow Photon Emission Rates [4] 


Source 

Emission 

477 I e 

Aurora," IBC I 

[OI] 32 5577 A 

1 kR 

II 

— 

10 kR 

III 

— 

100 kR 

IV 

— 

1000 kR 

Night airglow & 

[OI] 32 5577 A 

250 R 

(in the zenith) 

[Oik 6300 A 

50-100 R 


Na 5893 A 

— 


summer 

<30 R 


winter 

200 R 


Ha 6563 A 

5-20 R 


hy a 1215 A 

2.5 kR 


0 2 Atmospheric (0-1) 8645 A 

1.5 kR 


0 2 Herzberg (observable range) 

430 R 


OH (4-2) 1.58 /a 

175 kR 


OH (estimated total) 

4500 kR 

Twilight airglow c 

N 2 + 3914 A 

1 kR 

(referred to the zenith) 

(quiet magnetic conditions) 

— 


Na I 5893 A 

— 


summer 

1 kR 


winter 

5 kR 


[Oik 6300 A 

1 kR 


Ca II 3933 A 

150 R 


Li I 6708 A 

200 R 


[Nik 5199 A 

10 R 


0 2 IR Atmospheric (0-1) 1.58 /x 

20 kR 

Day airglow 

Na 5893 A 

— 

(referred to the zenith) 

summer 

2 kR 


winter 

15 kR 


[Oik 6300 A 

50 kR 


OI 8446 A 

0.5 kR 


OI 11,290 A 

0.5 kR 


N 2 + 3914 A 

f < 70 kR 
l> 1 kR 


"Recommended as definitions of the International Brightness Coefficients (IBC) [15,16]. 

6 Average values. 

c Approximate values of the maximum emission rates that are observed during twilight. These values are often 
governed by the time after sunset when observations first become possible. 
d Values predicted from theory [17-20]. 

e 4nI is the apparent emission rate in rayleighs.l R = an apparent emission rate of 1 megaphoton/cm 2 -sec (column). 


STELLAR RADIATION 


107 


Figure 5-20 shows the frequency distribution of air 
glow and weak auroral brightnesses near the geomag¬ 
netic pole (Thule, Greenland) and at a subauroral station 
(Fritz Peak, Colorado [9]. 

There is some evidence that suggests a general in¬ 
creasing brightness of airglow emissions toward higher 
latitudes and a bright belt at middle latitudes. 

5.4. Stellar Radiation 

5.4.1. Stellar Magnitudes [21]. The brightness of 
celestial bodies is usually measured in magnitudes. The 
scale of magnitudes is adjusted so that a star of magni¬ 
tude -fl.00 (first magnitude) gives a luminous flux of 
0.832 x 10~ 10 lumen cm -2 at a point outside the atmos¬ 
phere of the earth. 

The relation between the visible light received from 
two stars and their magnitudes is expressed by the formula 


logy = 0.4( m 2 — mi) 

i2 


where I = illuminance 


(5-1) 


w 

o 

z 

w 

ce 

K 

D 

O 

u 

o 

H 

z 

w 

u 

K 

w 

a, 



200 400 600 800 1000 
INTENSITY (rayleighs) 


Fig. 5-20. Frequency dis¬ 
tribution of airglow [9]. 


m — magnitude 

5.4.2. Stellar Spectral Classes. Under the Harvard system of classification the 
principal types of spectra are designated by the letters B, A, F, G, K, and M. Stars inter¬ 
mediate to these designations are designated by suffixed numbers from 0 to 9. 

The apparent temperatures corresponding to the various spectral classes are not 
always the same, but vary according to the methods used to measure or calculate the 
temperature. The following list should be considered only an approximation for main- 
sequence stars: 

Spectral Classification Surface Temperature of Star f°K) 


B -0 
A-0 
F-0 
G-0 
K -0 
M-0 


20,000 

11,000 

7.500 
6,000 
5,000 

3.500 


5.4.3. Numbers of Stars. Table 5-2 shows the estimated number of stars brighter 
than a given magnitude for both photographic and visual magnitudes. From mag¬ 
nitude 0 to 18.5, the figures are based on direct observation; the values from magnitude 
18.5 to 21 are extrapolated. 

The photographic results are based on all available material such as photographs, 
star charts, etc. The data for visual magnitudes are derived from the photographic 
results by allowing for the color of the stars. Very few stars are bluer than class A-0, 
for which class the visual and photographic magnitudes are equal; but many stars are 
redder and have color indices of +1 magnitude or more. A list of stars brighter vis¬ 
ually than the tenth magnitude, for example, will contain many red stars which are 
photographically of the eleventh magnitude or fainter, and a great many which are 
photographically fainter than the tenth magnitude. On the other hand, a list of stars 






108 


BACKGROUNDS 


to the tenth photographic magnitude will contain a few blue stars which are visually 
below the tenth magnitude, but not many. The difference in the numbers in the two 
columns is thus explained. As seen by the table, this effect increases for the fainter 
stars, which are generally redder than the brighter ones. Table 5-3 shows the percent¬ 
age of stars in the six principal spectral classes for various ranges of magnitudes. 


Table 5-2. Estimated Total Number of Stars 
Brighter Than a Given Magnitude [22] 


Photographic 

Magnitude 


_ Number of Stars _ 

Photographic Visual 


0 

— 

2 

1 

— 

2 

1 

— 

12 

2 

— 

40 

3 

— 

140 

4 

360 

530 

5 

1,030 

1,620 

6 

2,940 

4,850 

7 

8,200 

14,300 

8 

22,800 

41,000 

9 

62,000 

117,000 

10 

166,000 

324,000 

11 

431,000 

870,000 

12 

1,100,000 

2,270,000 

13 

2,720,000 

5,700,000 

14 

6,500,000 

13,800,000 

15 

15,000,000 

32,000,000 

16 

33,000,000 

71,000,000 

17 

70,000,000 

150,000,000 

18 

143,000,000 

296,000,000 

19 

275,000,000 

560,000,000 

20 

505,000,000 

1,000,000,000 

21 

890,000,000 

— 


Table 5-3. Percentage of Stars of 
Various Spectral Classes [22] 


Visual 

B-0 

B-8 

A- 5 

F-5 

G-5 

K-5 

Magnitude 

to B-5 

to A- 3 

to F- 2 

to G- 0 

to K- 2 

to M-8 

<2.24 

28 

28 

7 

10 

15 

12 

2.25 to 3.24 

25 

19 

10 

12 

22 

12 

3.25 to 4.24 

16 

22 

7 

12 

35 

8 

4.25 to 5.24 

9 

27 

12 

12 

30 

10 

5.25 to 6.24 

5 

38 

13 

10 

28 

6 

6.26 to 7.25 

5 

30 

11 

14 

32 

7 

7.26 to 8.25 

2 

26 

11 

16 

37 

7 

8.5 to 9.4 

2 

18 

13 

20 

36 

12 

9.5 to 10.4 

1 

16 

12 

24 

38 

9 

For All 

Magnitudes 

2 

29 

9 

21 

33 

6 

Photographic 

B-0 

B- 6 

A- 5 

F-5 

G-5 

K-5 

Magnitude 

to B-5 

to A- 4 

to F- 4 

to G-4 

toK- 4 

to M-8 

8.5 to 9.5 

2 

31 

16 

24 

24 

3 

9.5 to 10.5 

1 

24 

16 

31 

26 

3 

10.5 to 11.5 

1 

17 

13 

40 

27 

3 

11.5 to 12.5 

0 

10 

13 

47 

26 

3 

12.5 to 13.5 

0 

3 

10 

58 

26 

2 


The data are taken from the publications of the Harvard, McCormick, and Bergedorf 
Observatories. The discontinuity in trend appearing between the visual and photo¬ 
graphic groupings is in accordance with expectations. Of the stars brighter than 
magnitude 8.5, 99% belong to the six classes listed. 



STELLAR RADIATION 


109 


5.4.4. Galactic Concentration of Stars 

5.4.4.I. The Number of Stars (Galactic Concentration) in Different Parts of the Sky. 
Table 5-4 shows the number of stars per square degree brighter than a given photo¬ 
graphic magnitude, for different galactic latitudes. 


Table 5-4. Number of Stars Per Square Degree Brighter Than Photographic 
Magnitude as a Function of Galactic Latitudes [22] 


Photographic 

Magnitude 

+90° 

+40° 

+20° 

+10° 

0° 

1 

H- » 

o 

o 

o 

O 

1 

-40° 

-90° 

5.0 

0.014 

0.0175 

0.023 

0.031 

0.059 

0.045 

0.032 

0.0178 

0.012 

6.0 

0.039 

0.053 

0.071 

0.089 

0.166 

0.126 

0.087 

0.051 

0.042 

7.0 

0.015 

0.151 

0.20 

0.257 

0.436 

0.323 

0.224 

0.144 

0.123 

8.0 

0.275 

0.42 

0.59 

0.741 

1.230 

0.851 

0.617 

0.398 

0.316 

9.0 

0.724 

1.12 

1.62 

2.14 

3.55 

2.34 

1.69 

1.10 

0.832 

10.0 

1.78 

2.95 

4.50 

5.89 

10.5 

6.61 

4.68 

2.95 

2.09 

11.0 

4.3 

7.4 

12.0 

16.2 

30.9 

18.2 

12.8 

7.76 

5.25 

12.0 

10.2 

18.2 

32.0 

43.6 

89.1 

50.1 

34.7 

19.50 

13.2 

13.0 

24.0 

43.0 

79.0 

112.0 

245.0 

138.0 

89.1 

47.8 

30.2 

14.0 

50.0 

93.0 

190.0 

282.0 

661.0 

371.0 

218.0 

107.0 

60.3 

15.0 

95.0 

200.0 

457.0 

708.0 

1660.0 

977.0 

525.0 

218.0 

104.0 

16.0 

182.0 

407.0 

1047.0 

1778.0 

3981.0 

2455.0 

1175.0 

436.0 

182.0 

17.0 

338.0 

794.0 

2291.0 

4365.0 

9120.0 

5754.0 

2512.0 

832.0 

302.0 

18.0 

616.0 

1413.0 

4677.0 

9330.0 

20890.0 

12590.0 

4786.0 

1514.0 

501.0 

19.0 

770.0 

2180.0 

6860.0 

— 

— 

— — 

- 

- 

20.0 

21.0 

1670.0 

5000.0 

21200.0 



: : 

: 

““ 


5.4.4.2. Galactic Concentration of Stars of Various Spectral Classes. Table 5-5 
shows the average number of stars per 100 square degrees near the galactic equator 
and in regions remote from it for the six principal spectral classes. 

An approximation to the number of stars of a certain spectral class and magnitude 
range can be obtained by applying the data of Table 5-3 to Table 5-2, since Table 5-2 
gives the estimated number of stars brighter than a given magnitude for each magni¬ 
tude. For example, by interpolation of Table 5-2, the estimated number of stars 
brighter than magnitudes 7.25 and 8.25 may be obtained. By subtraction, the number 
of stars in the magnitude range 7.25 to 8.25 is obtained. The percentage of stars of 
the six principal spectral classes for this range of magnitudes, as shown in Table 5-3, 
can be used to obtain the approximate number of stars in these spectral classes for this 
range of magnitudes. 


Table 5-5. Galactic Concentration of Stars 
of the Principal Spectral Classes in 100 


Square Degrees Near 


Stellar 

Magnitudes 

Galactic 

Latitudes 

B 

A 

Above 7 m .O 


40°-90° 

0.2 

6.6 


0° 

10.8 

21.1 

7 m .O to 8 m .25 


40°-90° 

0.1 

6.6 


0° 

18.9 

75.8 


Galactic Equator [22] 


F 

G 

K 

M 

Total 

3.0 

3.4 

10.2 

1.5 

24.9 

5.1 

5.1 

15.1 

3.9 

61.1 

9.5 

16.4 

32.8 

6.1 

71.5 

13.6 

20.9 

53.9 

13.6 

196.7 


Table 5-6 gives more detailed information of the distribution of stars by spectral 
class and magnitude. There are differences in the data of Table 5-5 and 5-6 because 
somewhat different areas of the sky were considered in preparing the tables. For 
example, Table 5-5 considers the latitude from 40° to 90°, whereas Table 5-6 considers 


110 


BACKGROUNDS 


Table 5-6. Galactic Concentration of Stars 
of Various Spectral Classes [22] 


Spectrum 

Visual 

Magnitude 

B 

A 

Galactic Latitude 0° to 5° 

F G 

K 

M 

<6.0 

4.5 

6.0 

1.7 

2.1 

3.5 

1.3 

6.0 to 7.0 

6.3 

15 

3.4 

3.0 

12 

2.6 

7.0 to 8.25 

19 

76 

14 

21 

54 

14 

8.5 to 9.4 

46 

190 

85 

96 

200 

57 

9.5 to 10.4 

82 

610 

240 

310 

490 

150 

Photographic 

Magnitude 







9.5 to 10.5 

38 

510 

150 

220 

180 

19 

10.5 to 11.5 

87 

970 

430 

720 

460 

42 

11.5 to 12.5 

100 

1390 

1200 

1960 

940 

140 

Visual 

Magnitude 



Galactic Latitude 60° to 90° 



<6.0 

0.2 

2.6 

0.8 

1.0 

2.9 

0.7 

6.0 to 7.0 

0 

3.8 

1.8 

2.4 

7.5 

0.7 

7.0 to 8.25 

0 

7.4 

9.2 

16 

32 

6.3 

8.5 to 9.4 

0 

8 

20 

83 

75 

0 

9.5 to 10.4 

0 

8 

20 

170 

210 

16 

Photographic 

Magnitude 







9.5 to 10.5 

0 

9 

32 

120 

75 

9 

10.5 to 11.5 

0 

10 

27 

290 

160 

12 

11.5 to 12.5 

0.9 

14 

34 

680 

270 

26 


Note: The data are taken from the publications of the Harvard, 
McCormick, and Bergedorf Observatories. 


Table 5-7. 

Index of Apparent Galactic Concentration 

[22] 

Visual 

Magnitude 

b 

A 

F 

G 

K 

M 

<6.0 

22 

2.8 

2.3 

2.1 

1.2 

1.9 

6.0 to 7.0 

— 

4.0 

1.9 

1.2 

1.5 

3.7 

7.0 to 8.25 

— 

10 

1.5 

1.3 

1.7 

2.2 

8.5 to 9.4 

— 

24 

4.2 

1.2 

2.7 

— 

9.5 to 10.4 

Photographic 

Magnitude 


76 

12 

1.8 

2.3 

0.9 

9.5 to 10.5 

— 

56 

4.8 

1.8 

2.4 

2.1 

10.5 to 11.5 

— 

97 

16 

2.5 

2.9 

3.5 

11.5 to 12.5 

— 

99 

35 

2.9 

3.5 

5.5 


Note: The irregularities here are attributable in part to inadequate sampling. 


the latitude from 60° to 90°, in arriving at an average galactic distribution. The most 
important difference is that Table 5-6 has been prepared by selecting narrower ranges 
of stellar magnitude. 

Data of galactic distribution are not presented for stars of magnitudes less than 5 
because the total number of these stars is not large enough to make the concept of the 
number of stars per square degree meaningful. 

Table 5-7, an index of apparent galactic concentration, has been prepared from Table 
5-6 by taking the ratios of numbers of stars in low latitudes to the numbers in high 
latitudes. For a given spectral class, more stars are concentrated in the lower galactic 
latitudes as the index number becomes higher. 

5.4.5. Spectral Distribution of Stellar Radiation.* Figure 5-21 shows relative 
spectral distribution of stellar radiation as a function of star classes and surface tem¬ 
perature. The family of curves in Fig. 5-22 shows absolute spectral distribution of 


*See also Notes Added in Proof, page 171. 



STELLAR RADIATION 


111 



TEMPERATURE, Thousands of Degrees K 

I-1—I-1-1-1 

M. K. G. F a A„ B 


CLASS OF STAR 

Fig. 5-21. Relative spectral distribution of stellar 
radiation as a function of star classes [22]. 



i-1— 

M o K o 


~r 

G„ 


T" 

F„ 


T" 


CLASS OF STAR 

Fig. 5-22. Absolute spectral distribution of stellar radiation [22]. 


B„ 


stellar radiation. In this figure, the absolute magnitude of the radiant energy falling 
below a specified wavelength is plotted as a function of the surface temperature of 
the stars. Further, the curves have been normalized so that the amount of energy 
in the visible region is constant for all the stars of any given magnitude. This value 
is represented by one vertical division of the graphic scale on Fig. 5-22. 




























































112 


BACKGROUNDS 


5.4.6. Determining Spectral Irradiance of Celestial Bodies [23]. The following 
data and methods permit determining with reasonable accuracy the spectral irradiance 
values of the brightest stars and planets in the infrared region. These data have been 
calculated from published measurements of visible irradiance and effective temperature, 
and include the complete spectral region from 0.1 to 100 /x. 

The data used pertain to irradiance received above the atmosphere. Values for 
absorption by the atmosphere in the various spectral regions can be readily applied to 
the chart values. 

Table 5-8 [21] shows the visible magnitude and effective temperature ( T e ff= W/cr) 
values for the brightest celestial bodies and also for the important "red stars.” The 
list contains all the stars in Schlessinger’s Catalogue of Bright Stars which give an 
irradiance of at least 10 -12 w cm -2 in either the PbS region (1-3 g) or the bolometer 
region (0.3-13.5 g). Equation (5-2) is plotted in Fig. 5-23. 

f J\(T)Se\ d\ 

T)e (T) = - ( 5 - 2 ) 

Jx(T)dk 

Jo 

is the fraction of total radiation emitted by a blackbody at some temper¬ 
ature T, visible to the standard observer 

is the ordinate of the Planck blackbody radiation curve at wavelength A 
and temperature T 

is the fractional response of the eye at the same wavelength. 


where r\ e {T) 
J\(T) 
S P \ 


Table 5-8. Visual Magnitudes and Effective Temperature 
of Planets and the Brightest Visual and Red Stars [21] 



Name 

Visual Magnitude 

(m„) 

Effective Temperature 
T (°K) 

1 . 

Moon (full) 

(Planets) 

-12.2 

5,900 

2. 

Venus (at brightest) 

-4.28 

5,900 

3. 

Mars (at brightest) 

-2.25 

5,900 

4. 

Jupiter (at brightest) 

-2.25 

5,900 

5. 

Mercury (at brightest) 

-1.8 

5,900 

6. 

Saturn (at brightest) 

(Stars) 

-0.93 

5,900 

1 . 

Sirius 

-1.60 

11,200 

2. 

Canopus 

-0.82 

6,200 

3. 

Rigel Kent (double) 

0.01 

4,700 

4. 

Vega 

0.14 

11,200 

5. 

Capella 

0.21 

4,700 

6. 

Arcturus 

0.24 

3,750 

7. 

Rigel 

0.34 

13,000 

8. 

Procyon 

0.48 

5,450 

9. 

Achernar 

0.60 

15,000 

10. 

/3 Centauri 

0.86 

23,000 

11. 

Altair 

0.89 

7,500 

12. 

Betelguex (variable) 

0.92 

2,810 

13. 

Aldebaran 

1.06 

3,130 

14. 

Pollux 

1.21 

3,750 

15. 

Antares 

1.22 

2,900 

16. 

a Crucis 

1.61 

2,810 

17. 

Mira (variable) 

1.70 

2,390 

18. 

/3 Gruis 

2.24 

2,810 

19. 

R. Hydrae (variable) 

3.60 

2,250 



STELLAR RADIATION 


113 


Fig. 5-23. Fraction of the 
total radiation emitted by 
a blackbody at temperature 
T, visible to the standard 
observer [21]. 



After 7) e (T) is found, the stellar magnitude m, of the body may be used to obtain 
the total blackbody spectral irradiance, as follows: 


m v 


= 2.5 logio 


/(m t ,) 

Io 


(5-3) 


At the top of the atmosphere, zero visible magnitude corresponds to a visible irradiance, 
Io, of 3.1 X 10~ 13 w/cm 2 . The value for I(m r ), for any quoted value of stellar magnitude, 
may then be obtained by the solution of Eq. (5-3). This function is plotted in Fig. 5-24. 



IRRADIANCE (w cm" 2 ) 


Fig. 5-24. Effective irradiance in the visible-region (standard observer) 
versus visual magnitude [21]. 






114 


BACKGROUNDS 



Fig. 5-25. Peak spectral irradiance from values of visual magnitude 
and effective temperature or spectral class [21]. 


The irradiance received over the total wavelength spectrum at the top of the atmos¬ 
phere is therefore the quantity 


I(m v ) 

r}e(T) 

and once the value of peak spectral irradiance is determined, the shape of the spectral 
irradiance curve follows the Planck radiation function. 

The peak irradiance is 


H 


X peak 


Iim r) 

T le ( T ) 


W K 


max 



(5-4) 


where W\ max is the maximum value of the Planck function, and equals 1.290 X lO -15 ? 15 
w cm -2 /a -1 . Eq. (5-4) then becomes 

Hw = x 2 - 272T x 10 “ w cm ‘ 2 <5 ‘ 5) 

Equation (5-5) evaluated and plotted as a function of T for various values of magnitude 
m v is shown in Fig. 5-25. This graph can be used to find the peak spectral radiance, 
H x peak , for any values of T and m v - Using Fig. 5-25 and Wien’s law, the spectral ir¬ 
radiance curves for any star or planet may be obtained. (In determining the spectral 






THE EARTH AS A BACKGROUND 


115 


irradiance of the planets, an effective temperature of 5900°K was assumed.) The 
shape of all these irradiance curves are identical; they are blackbody curves normalized 
to their peak value. 

5.5 The Earth as a Background* 

5.5.1. Geometric Relationships. Figures 5-26, 5-27, and 5-28 present some impor¬ 
tant relationships bearing on satellite viewing of the earth. 



Fig. 5-26. Ranges and view angles [3]. 


ALTITUDE, y (degrees) 



• H 

s 


H 

O 

z 

< 

OS 

Eh 

Z 

< 

A 

w 


Fig. 5-27. View angles for 200 n mi orbit [3]. 


3 

£ 




z 

o 

N 

OS 

o 

OS 

o 

w 

_) 

o 

z 

< 

z 

o 

Eh 

< 

> 

w 

A 

w 


to 

o> 

0 ) 

Sh 

fafi 

0> 

T3 



CO 

<u 

<u 

S-, 

to 

o 

X} 


K 



u 

OS 

< 

W 

J 

U 

OS 

u 

Eh 

< 

u 

OS 

o 


»-0 


*See also Notes Added in Proof, page 171. 











BACKGROUNDS 


116 

In Fig. 5-28, point P (as an example) represents a vehicle 400 n mi high at an elevation 
of 35°. The slant range is 655 n mi, and the great circle arc (angle between vectors 
located at the center of the earth and pointing respectively to the satellite and to the 
ground point viewed) is 7°. 



Fig. 5-28. Satellite coordinate conversion [3]. 


5.5.2. Path Lengths. Refer to Fig. 5-29. The length of a line between any two 
points at different altitudes is found according to the following general equations: 

(Re “I - C) 2 = (R £+ A) 2 + L 2 + 2 L(Re + A) cos 8 = (i?£ + A) 2 + L 2 + 2 L(Re 4- A) sin y 

L z — (i?£+ C) 2 — (R £+ A) 2 — 2 L(R £-f A) sin y 

L = y/(R E + C) 2 - (R e +A) 2 cos 2 y - (R E + A) sin y (5-6) 

where R E = radius of sphere 

A = altitude of the background point 
C — altitude of the observer 

y = elevation angle of the background point position (from local horizontal) 
a = elevation angle from nadir at observer’s point. 

The angle a is computed by the following relationship: 

, (R C) 2 + L 2 — (R E -\- A) 2 


a = cos 


2 (R e +C)L 


(5-7) 






































THE EARTH AS A BACKGROUND 


117 


Observer 



Figure 5-30 illustrates the scattering angle (3, the sun’s elevation angle y, the satellite 
scanner’s elevation angle a, and the scanner’s azimuth angle 4> from the direction of 
the sun. 

The scattering angle is: 

f3 = cos -1 (cos y sin a cos </> — sin y cos a) (5-8) 

Figure 5-31 illustrates these angles for a spherical earth. Equation (5-8) becomes 
/3 = cos -1 [cos (90° — (£ — A)] sin a cos </> — sin [90° — (£ — A)] (5-9) 


Sun 



Sun 



Fig. 5-30. Scattering-angle geometry [3]. Fig. 5-31. Solar scattering angle [3]. 














118 


BACKGROUNDS 


Sun's Position Sun's Altitude at Noon 

Equinox (90° - Lat.) 

Summer Solstice (90° - Lat.) + 231/2° 

Winter Solstice (90° - Lat.) - 23 1/2° 



DAY OF YEAR 

Fig. 5-32. Solar declination to equator [3]. 

where £ is the elevation angle (latitude) of the observed point from the earth’s equatorial 
plane, and A is the latitude of the sun’s prime ray. Figure 5-32 shows the range of 
the sun’s declination angle. (Both £ and A are taken positive for a North latitude.) 

5.6. Cloud Meteorology 

Clouds are classified into ten main groups called genera. These are cirrus, cirro- 
cumulus, cirrostratus, altocumulus, altostratus, nimbostratus, stratocumulus, stratus, 
cumulus, and cumulonimbus. 

The part of the atmosphere in which clouds are usually present is divided into three 
regions. Each region is defined by the range of levels at which clouds of certain genera 
occur most frequently. 

(а) High-level clouds —cirrus, cirrocumulus, and cirrostratus 

(б) Middle-level clouds—altocumulus 

(c) Low-level clouds — stratocumulus and stratus 

The regions overlap, and their limits vary with latitude. Their approximate ranges 
are shown in Table 5-9. Figures 5-33 to 5-38 show the mean cloudiness in percentage 
of sky cover throughout the world for various months of the year. 


Table 5-9. Definition of Cloud State Altitudes [3] 


Cloud 

Level 

Polar Regions 

Temperate Regions 

Tropical Regions 

High 

3-8 km 

(10,000-25,000 ft) 

5-13 km 

(16,500-45,000 ft) 

6-18 km 

(20,000-60,000 ft) 

Middle 

2-4 km 

(6500-13,000 ft) 

2-7 km 

(6500-23,000 ft) 

2-8 km 

(6500-25,000 ft) 

Low 

Earth’s surface 
to 2 km 
(6500 ft) 

Earth’s surface 
to 2 km 
(6500 ft) 

Earth’s surface 
to 2 km 
(6500 ft) 







CLOUD METEOROLOGY 


119 



170W 110W 50W 10E 70E 130E 170W 


Fig. 5-33. Mean cloudiness in percentage of sky cover, month of January [25]. 


170W110W50W 10E 70E 130E 170W 



170W 110W 50W 10E 70E 130E 170W 


Fig. 5-34. Mean cloudiness in percentage of sky cover, month of March [25]. 



















































































120 


BACKGROUNDS 



170W HOW 50W 10E 70E 130E 170W 


170W110W50W 10E 70E 130E 170W 


Fig. 5-35. Mean cloudiness in percentage of sky cover, month of May [25]. 



170W 110W 50W 10E 70E 130E 170W 


170W110W50W 10E 70E 130E 170W 


Fig. 5-36. Mean cloudiness in percentage of sky cover, month of July [25]. 














































































CLOUD METEOROLOGY 


121 



170W 110W 50W 10E 70E 130E 170W 

Fig. 5-37. Mean cloudiness in percentage of sky cover, month of September [25]. 


170W110W50W 10E 70E 130E 170W 



170W 110W 50W 10E 70E 130E 170W 


Fig. 5-38. Mean cloudiness in percentage of sky cover, month of November [25]. 













































































122 


BACKGROUNDS 


5.6.1. Cirrus Clouds. The tropopause represents the upper limit of the cloud 
atmosphere. The highest clouds appearing within the troposphere are composed of 
large ice crystals of about 100 (jl. Frequently these particles become oriented in the 
same direction, giving rise to unusual visible, and possibly infrared, effects such as 
haloes and arcs. 

Tropopause and cloud top statistics are not available for the central Eurasian land 
mass. Cirrus height observations have not been reported anywhere north of 55° lati¬ 
tude. Inferences can be made about the annual tropopause distribution over Eurasia, 
and from this a cirrus top height model constructed. The correlation between the 
two parameters is based on American statistics. Between 50° and 70° N it is expect¬ 
ed that 90% of the annual clouds will be below 32,000 ft, and 99% will be below 36,000 
ft. 

Figure 5-39 shows cloud top and tropopause heights based on a collation of cirrus 
and tropopause data averaged on a yearly basis for the entire United States. Figures 
5-40 and 5-41 represent the distribution of tropopause and cloud heights between 50° 
and 90° N latitude. 

Based on deductions from Asian climatology, a crude time-frequency occurrence 
chart has been estimated and is shown in Fig. 5-42. Averaging the entire year be¬ 
tween 50° and 70° N, cirrus clouds are expected 35% of the time. This means that 
cirrus will be encountered 1% of the time above 34,000 ft, and 10% of the time above 
30,000 ft. 



Fig. 5-39. Distribution of cloud top and tropopause heights, 
United States average [3]. 




















CLOUD METEOROLOGY 


123 


Fig. 5-40. Distribution of tro- 
popause and cloud tops, 50° to 
70° N latitude, Eurasian average 
[3]. 




Fig. 5-41. Distribution of tro- 
popause and cloud tops, 65° to 
90° N latitude, Eurasian average 
[3], 


ALTITUDE 


































































124 


BACKGROUNDS 



Fig. 5-42. Estimated annual temporal frequency of 
cirriform clouds. Dotted portion delineates area where 
20 or more thunderstorms per year are reported. 
Overall average = 35% [3]. 


5.6.2. Stratospheric Clouds [3]. Two types of clouds appear in the upper strato¬ 
sphere: nacreous clouds at an average height of 24 km, and noctilucent clouds at a 
height of about 82 km. 

Nacreous clouds appear rarely and then mainly in high latitudes characterized by 
mountainous terrain. They are generally observed in the direction of the sun during 
sunset or sunrise and are irridescent. Characteristic synoptic conditions that exist 
with these clouds are strong and consistent northwest winds extending to great heights 
with below average stratospheric temperatures. Theoretical considerations of water- 
droplet and ice-crystal growth in nacreous clouds suggest that the radii are less than 
1.2 /x, with a very narrow size spectrum of about 0.1 /x. The particle concentration 
should be essentially that of the available condensation nuclei, about 1/cm 3 . The 
liquid water content would be therefore between 10 -12 to 10“ 11 g/cm 3 . Such liquid 
water content is lower by about a factor of 10 4 than those observed in the tropospheric 
clouds. 

Noctilucent clouds are visible against the nighttime sky when the upper levels of 
the atmosphere are still illuminated by sunlight. These clouds have generally been 
reported only in the Northern Hemisphere during summer (August through October) 
within a restricted zone of latitudes extending from about 45° to 63° N. 

Sunlight scattered from noctilucent clouds exhibits a spectrum and a degree of 
polarization which can be attributed to the scattering of sunlight by dielectric par¬ 
ticles with predominant radii of around 0.1 /x and not greater than 0.2 to 0.4 /x. The 
observed brightness of the clouds suggests that the corresponding concentrations and 
matter content should be between 1 and 10 -2 particles/cm 3 and between 10“ 17 and 
10 _ 10 g/cm 3 , respectively. Such cloud-particle concentrations are about five orders 
of magnitude less than those given for nacreous clouds. 

5.6.3. Probability of Coverage at Various Altitudes. Figures 5-43 through 5-58 
are charts showing, for the Northern Hemisphere, altitudes above which the proba¬ 
bilities of less than 0.1 sky coverage are 95, 90, 80, and 60 percent. Charts are pre¬ 
sented for the midseason months January, April, July, and October. The criterion of 
less than 0.1 sky cover (actually less than 0.05 sky cover) can be taken as essentially no 
interference by clouds for air-to-air operation. 







CLOUD METEOROLOGY 


125 



Fig. 5-43. Altitudes (thousands of feet MSL) above which there is 95% probability 
of having less than 0.1 sky cover, month of January [26]. 
















126 


BACKGROUNDS 



Fig. 5-44. Altitudes (thousands of feet MSL) above which there is 90% probability 
of having less than 0.1 sky cover, month of January [26]. 














CLOUD METEOROLOGY 


127 



Fig. 5-45. Altitudes (thousands of feet MSL) above which there is 80% probability 
of having less than 0.1 sky cover, month of January [26]. 
















128 


BACKGROUNDS 



Fig. 5-46. Altitudes (thousands of feet MSL) above which there is 60% probability of having 
less than 0.1 sky cover, month of January [26]. 
















CLOUD METEOROLOGY 


129 



Fig. 5-47. Altitudes (thousands of feet MSL) above which there is 95% probability of having 
less than 0.1 sky cover, month of April [26]. 



























02 - 


130 


BACKGROUNDS 



Fig. 5-48. Altitudes (thousands of feet MSL) above which there is 90% probability of having 
less than 0.1 sky cover, month of April [26]. 



















CLOUD METEOROLOGY 


131 



Fig. 5-49. Altitudes (thousands of feet MSL) above which there is 80% probability of having 
less than 0.1 sky cover, month of April [26]. 

















132 


BACKGROUNDS 



Fig. 5-50. Altitudes (thousands of feet MSL) above which there is 60% probability of having 
less than 0.1 sky cover, month of April [26]. 





















CLOUD METEOROLOGY 


133 



Fig. 5-51. Altitudes (thousands of feet MSL) above which there is 95% probability of having 
less than 0.1 sky cover, month of July [26]. 





















134 


BACKGROUNDS 



Fig. 5-52. Altitudes (thousands of feet MSL) above which there is 90% probability of having 
less than 0.1 sky cover, month of July [26]. 























CLOUD METEOROLOGY 


135 



Fig. 5-53. Altitudes (thousands of feet MSL) above which there is 80% probability of having 
less than 0.1 sky cover, month of July [26]. 
























136 


BACKGROUNDS 



Fig. 5-54. Altitudes (thousands of feet MSL) above which there is 60% probability of having 
less than 0.1 sky cover, month of July [26]. 













CLOUD METEOROLOGY 


137 



















138 


BACKGROUNDS 



Fig. 5-56. Altitudes (thousands of feet MSL) above which there is 90% probability of having 
less than 0.1 sky cover, month of October [26]. 
























CLOUD METEOROLOGY 


139 



Fig. 5-57. Altitudes (thousands of feet MSL) above which there is 90% probability of having 
less than 0.1 sky cover, month of October [26]. 





















140 


BACKGROUNDS 



Fig. 5-58. Altitudes (thousands of feet MSL) above which there is 60% probability of having 
less than 0.1 sky cover, month of October [26]. 
























STRATOSPHERIC AEROSOLS 


141 


5.7. Stratospheric Aerosols [3, 63-68] 

A uniform distribution of the stratospheric aerosol content tends to decrease infrared 
gradients. Apparently the stratosphere will contribute to background noise because 
of the tendency of particles to form clouds which become arranged in periodic struc¬ 
tures. Intensities will be high at small scattering angles, and atmospheric attenua¬ 
tion is negligible at these high altitudes. Table 5-10 presents a summary of informa¬ 
tion on the particle content of the stratosphere. 


Table 5-10. Particle Content of the Stratosphere [3] 


Altitude 

Concentrations 

Radii 

(km) 

(no. per cm 3 ) 

(/a) 

10-30 

10- 1 to 1 

>0.08 

or 

-0.10 

10-30 

10“ 2 to 10 1 

-0.15 

10-30 

~<10 3 

-0.8 

(horizontal 

orientation) 

17-31 

~<1 

-1.5 

30-80 

1 

-0.1 

(assumed) 

74-92 

10~ 2 

-0.1 

80 

10“ 4 to 10- 1 

0.1 

(assumed) 


Typical Band 

Remarks Spacings 

(km) 

Stable dust layer, — 

17 to 22 km 

— sulfur 

Stable dust layer, — 

17 to 22 km 

— sulfur 

Temporary layers of 1 

volcanic pumice 

Nacreous clouds 40 

consisting of 
ice crystals 

Theoretical by — 

measurements of 
conductivity 

Noctilucent cloud 10 and 60 

(dust layer or 
ice crystal) 

Theoretical inter- — 

planetary dust 
sources, 10“ 23 to 
10“ 20 g/cm 3 


Stratospheric particulate matter may be divided into two classes: dust particles 
and condensed water. 

Catastrophic volcanic eruptions and forest fires have deposited vast quantities of 
dust in the upper atmosphere; these can indirectly increase the upward intensity of 
reflected sunlight by acting as nucleating agents for ice. Unusual concentrations of 
clouds might result. 

The earth is surrounded by belts of dust, smoke, and ice particles. Encounters with 
the dust by the earth’s gravitational field causes an accretion of 10 to 50 lb of matter 
per square mile per year, based on an estimate of 24,000,000 visible meteors per day [25]. 
Some of this dust is concentrated into two extreme outer shells: the lighter smoke 
between altitudes of 2000 to 4000 mi, and dust from 600 to 1000 mi. 

In general, the total particle concentrations just above the tropopause are between 
10 and 100/cm 3 , but decrease to 1/cm 3 or less above about 20 km. 


142 


BACKGROUNDS 


The manner in which a particle scatters light depends on the ratio of its radius to 
the wavelength of light. For ratios up to about 0.08, Rayleigh s laws hold; between 
0.08 and 3 the Mie theory is used; and at larger values, the laws of geometrical optics 
are satisfactory. Figure 5-59 gives examples of particle sizes. Figure 5-60 shows 
the concentration of different particle sizes at various altitudes. 


os 

w 

H 

w 

s 

2 

p 


10,000 

1,000 

. 100 

L 

10 
1 



Heavy Industrial Dust 


mmmmmmmmmmm Smallest Easily Visible Dirt 
Paint Pigments 
Bacteria, Pollen 



0.1 - 

0.01 ■ 

0.001 
0.0001 1 

Fig. 5-59. Examples of particle sizes [3]. 


Tobacco Smoke 
Fine Carbon Black 

Gas Molecules 


Fig. 5-60. Vertical profiles of particle concentrations. 
Curve A = radii greater than 0.08 p,; curve B = radii 
greater than 0.1-0.3 /a; curve C = Aitken nuclei (0.01- 
to 0.1-p. radii) [3], 


£ 

x 

u 

Q 

P 

H 

►-H 

P 

< 


30 

25 

20 

15 

10 

5 

0 


“ \ 

- \ 






V. 

- B , 


\a 


! 

2 

1 

1 

/ 


6 




/ 

1 

t 

/ 


c 

h Tr 

opo-- 

ausel 





5 









10' 3 10' 2 10' 1 1 10 10 2 10 3 
CONCENTRATION (No./cm 3 ) 


5.8. Spectral Radiance of Terrain 

The apparent spectral radiances discussed in this section include thermal emission 
and radiation reflected by the ground, as well as scattered and emitted radiation con¬ 
tributed by the atmosphere in the line of sight. 

5.8.1. Terrain Temperature. Soil or other terrestrial surfaces have a mean tem¬ 
perature value of approximately 300° K, and peak radiance is near 10 ix. The earth’s 
surface temperature depends upon the incident solar radiation and the radiative 
boundary conditions as well as conductive and convective processes. The latter proc¬ 
esses depend on the physical and chemical characteristics of particular components 
of the terrain and the local weather conditions. 

5.8.2. Terrain Emissivity and Reflectivity.* The amount of radiation that is ab¬ 
sorbed, reflected, or scattered varies with wavelength and with the nature of the terrain. 

The reflectance values for natural objects, at wavelengths shorter than 3 p,, range 
from 0.03 for bare ground or ocean to 0.95 for fresh snow [1,11,24,27,28,291. In the long- 
wavelength region reflectance values range from practically zero to 0.72 [30-321. 


*See also Notes Added in Proof, page 171. 



























SPECTRAL RADIANCE OF TERRAIN 


143 


5.8.3. Spectra in the Emission Region. Figure 5-61 shows the comparative 
spectral radiances of a patch of ground at an airfield (Colorado) observed on a clear 
night, and during the following morning with the sun shining on it. 

Figure 5-62 shows the radiance of the night sky just above the horizon and that of 
the ground at the same angle below the horizon. 



Fig. 5-61. Day and night radiances 
of grass-covered field (Peterson Field, 
Colorado) [33]. 




Fig. 5-62. Comparative spectra of the 
ground and sky near the horizon (Peterson 
Field, Colorado) [33]. 


Fig. 5-63. Radiance of an urban area and 
of clear zenith sky (Colorado Springs from 
Pikes Peak) [33]. 


The spectra of distant terrain do not always conform to the blackbody characteris¬ 
tics observed in the radiance of nearby terrain. This can be seen in Fig. 5-63, where 
the upper curve represents the radiance of a city (Colorado Springs) on a plain as 
viewed from the summit of a mountain (Pikes Peak) at a distance of about 15 mi. The 
situation illustrated in Fig. 5-63 is, in a sense, the reverse of that shown in Fig. 5-7 
where a cooler cloud was seen from a lower and warmer environment. 











144 


BACKGROUNDS 


Figure 5-64 shows the diurnal variation in the 10-/* radiance of selected backgrounds 
on the plains, measured from the summit of a mountain (Pikes Peak). The line-of- 
sight distances are: forest, 30 mi; grassy plains, 21 mi; airfield, 19 mi; city, 15 mi. 

Figure 5-65 shows the spectral radiance of dry sand (Cocoa Beach, Florida). The 
9-/* dips in the reflectance of the sand for curves A and C correspond with a wave¬ 
length of relative poor emissivity. The reason is that the crystals of common silica 
sand exhibit reststrahlen at 9 /*. With overcast sky (curve B ), the added sky radiance 
reflected at this wavelength just compensates for the loss of emissivity. 

The effect of moisture on the radiance of sand is seen in Fig. 5-66. 



TIME 

Fig. 5-64. Diurnal variation in the 10-/x radiance of selected backgrounds 
[33]. SS = sunset; SR = sunrise; ENT = end of nautical twilight; BNT = be¬ 
ginning of nautical twilight. 



Fig. 5-65. Spectral radiance of dry sand 
(Cocoa Beach, Florida) [33]. A = sunlit 
sand, B = sand under a cloudy night sky, C = 
sand on a clear night. 



Fig. 5-66. Spectral radiance of moist sand, 
(Cocoa Beach, Florida); A = dry sand, B = 
extremely wet sand, C = moist sand [33]. 






SPECTRAL RADIANCE OF TERRAIN 


145 


5.8.4. Spectra in the Scattering Region. The daytime spectra of objects at ambient 
temperatures show minima around 3 to 4 /x. In the more transparent regions of the 
spectrum between 3 and 5 /x, in the daytime, the sky a few degrees above the horizon 
radiates less than the ground a few degrees below. In the scattering region of the 
spectrum, the sky and the ground often show radiances of comparable magnitude. 
Usually the ground radiance is somewhat higher than that of the sky, and is frequently 
a minimum near the horizon. 

Figures 5-67 and 5-68 are elevation scans of the spectral radiance near the horizon 
at different wavelengths. 

The scan covers alternating patches of shaded and sunlit ground, trees, mountains 
(northern slope of Pikes Peak, Colorado), and the clear sky up to approximately 15° 
or 20° near the horizon. 


Clear Afternoon 
Azimuth 165° 



Fig. 5-68. Elevation scans across terrain 
at the same time as in Fig. 5-67 at fixed 
wavelengths in the intervals 3.5 to 5.2 /x 
[33]. 


Fig. 5-67. Elevation scans across mountainous 
terrain at fixed wavelengths in the interval 1.8 to 
3.2 /x [33]. 



5.8.5. Spectral Radiance of Various Objects and Surfaces [31,34,35]. Figures 
5-69 through 5-88 are measurements of the infrared spectral radiance from various ob¬ 
jects and surfaces. These measurements were made under different temperatures, 
humidity, sky conditions, etc. In the figures, a H and a v are the angular fields of view in 
the horizontal and vertical directions in object space. 







SPECTRAL RADIANCE (w 


146 


BACKGROUNDS 



Fig. 5-69. Daytime spectral radiance of miscellaneous targets [31]. 





























SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


147 



Fig. 5-70. Spectral radiance of concrete, winter day, clear [34]. 






















SPECTRAL RADIANCE (w 


148 


BACKGROUNDS 



















SPECTRAL POWER DENSITY AT COLLECTOR" (w cm 


SPECTRAL RADIANCE OF TERRAIN 


149 


Field of View 


-8 x 10 



-2 x 10 


-3 x 10 


-4 x 10 


3 

Run 

Ambient 

4 

Chopper 

5 

WAVELENGTH (p) 


Temp. 

Temp. 

Illumination 

Time 

Weather 

© 

25°C 

28.8°C 

Shade 

1:00 P.M. 

Sunny, Clear 

■ 

29.2°C 

30.8°C 

Sun 

3:30 P.M. 

Sunny, Clear 

▲ 

27.4°C 

30.6°C 

Sun 

5:20 P.M. 

Sunny, Clear 

X 

27°C 

30.8°C 

Sun 

6:25 P.M. 

Sunny, Clear 

I 

27.8°C 

30.9°C 

Sun 

7:15 P.M. 

Sunny, Clear 

& 

24°C 

29.7°C 

Shade & 
Partial Sun 

8:05 P.M. 

Nearing 

Sundown 

Clear 


Fig. 5-72. Spectral radiance of concrete wall during an afternoon 
[34]. Reference temperatures = chopper temperatures. 



































































SPECTRAL RADIANCE (w 


150 


BACKGROUNDS 



Fig. 5-73. Spectral radiance of concrete, summer day, overcast [34], 



















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


151 
















SPECTRAL RADIANCE (w 


152 


BACKGROUNDS 




















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


153 



Fig. 5-76. Spectral radiance of concrete at night [34). 




















SPECTRAL RADIANCE (w 


154 


BACKGROUNDS 



Fig. 5-77. Spectral radiance of damp concrete [34], 

















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


155 



Fig. 5-78. Spectral radiance of concrete through falling snow [34], 

























SPECTRAL RADIANCE (w • cm • sr 


156 


BACKGROUNDS 



Fig. 5-79. Spectral radiance of concrete wall from four different angles [34], 

































SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


157 

























SPECTRAL RADIANCE (w 


158 


BACKGROUNDS 





















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


159 


JO' 3 

H 

1 

a. 

H 

Sh 

CO 

CSI 

1 

6 4 

cj - 4 

10 
js 

w 

u 

z 

< 

Q 

s 

s 

H 

U 

W 

ex 

M -5 

10 b 

IQ' 6 

Refe 

rence T = 0 

°K 

Fielc 

ff H = 3 - 

°y = 

Range ~ 

l of View 

14 x10' 3 r 
85 x10" 4 r 
300 ft 

ad 

ad 























3 1 2 

< 

5 6 


WAVELENGTH (jll) 

Fig. 5-82. Spectral radiance of grass, summer day, clear [34]. 


















SPECTRAL RADIANCE (w 


160 


BACKGROUNDS 




















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


161 


i 




t-c 

CO 


CM 


I 


s 

O 


Field of View 



WAVELENGTH (/i) 


Fig. 5-84. Spectral radiance of grass, summer night, clear [34]. 






















162 


BACKGROUNDS 




















SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


163 



Fig. 5-86. Spectral radiance of snow cover under various weather 
conditions at various ambient temperatures [34], Note the high 
intensity of scattered sunlight. 


























SPECTRAL RADIANCE (w 


164 


BACKGROUNDS 



























SPECTRAL RADIANCE (w 


SPECTRAL RADIANCE OF TERRAIN 


165 



Fig. 5-88. Spectral radiance of sky, concrete, snow, and grass, winter night [34]. 






















166 


BACKGROUNDS 


5.9. Marine Backgrounds 

The radiance of the sea surface at night is the sum of its thermal emission and re¬ 
flected sky radiance. Factors that determine the character of the marine background 
are: 

1. The infrared optical properties of sea water. 

2. Sea-surface geometry and wave-slope distribution. 

3. Sea-surface temperature distribution. 

Atmospheric transmission and emission in the optical path from scene to observing 
instrument is covered in Chapter 6. 

5.9.1. Infrared Optical Properties of Sea Water. Water is essentially opaque 
to infrared radiation longer than 3 fx. Few liquids have absorption coefficients of 
the same order of magnitude. Consequently, the sea surface, which is 0.01 cm thick, 
determines the radiance of the sea. Subsurface scattering of sky radiation is absent. 
The optical influence of thin layers of surface contamination is negligible except for 
the suppression of capillary waves by surface tension changes — causing "slicks.” 
There is no significant difference in the transmissivity of sea and distilled water for 
these thin layers in the 2- to 15 -/li region. 

The infrared transmissivity, reflectivity, emissivity and indices of refraction for 
water are shown in Figs. 5-89 to 5-92 [36,37]. 



Fig. 5-89. Transmissivity of 0.003 cm of sea water and reflectivity of a free sea-water surface 
(dashed line) [36], 


REFLECTIVITY 








MARINE BACKGROUNDS 


167 



Fig. 5-90. Indices of refraction of water calculated from reflectivity 
data in Fig. 5-89. 



Fig. 5-91. Reflection from a water surface at 0°, 60°, and 80° angle 
of incidence calculated from data in Fig. 5-90. 



Fig. 5-92. Reflectivity and emissivity of water (2- to 15-/n average) versus angle 
of incidence, calculated from averaged data of Fig. 5-90; see [37]. Note scale 
change. 













168 


BACKGROUNDS 


5.9.2. Sea-Surface Geometry. The effect of wave slope on the reflectivity of a sea 
surface roughened by a Beaufort 4 wind (11 to 16 knots, white caps) is seen in Fig. 5-93. 
Here, for an average rough sea, the reflectivity approaches 20% near the horizon. 
Consequently, the emissivity remains at 80% or higher. 

The radiance of the sea surface along an azimuth 90° from that of the sun, in day¬ 
light for clear and for overcast conditions, is shown in Fig. 5-94. 

Information is lacking on similar observations for the radiance of the sea surface 
at night. However, the variation of sky radiance with zenith angle is similar day 
and night, and the photographic reflectivity is about equal to the average for the in¬ 
frared from 2 to 15 /Lt (Fig. 5-92). Consequently, the curves in Fig. 5-94 are instructive 
because they show the general shape of that part of the radiance of the sea surface 
at night due to the reflection of sky radiation. To these curves must be added the 
infrared radiance of the sea surface due to its temperature. 


Fig. 5-93. Reflection of solar radiation from a 
flat surface (cr — 0) and a surface roughened by a 
Beaufort 4 wind (cr = 0.2). The albedo R varies 
from 0.02 for a zenith sun, i/»(= 0°) to unity for 
the sun at the horizon (t// = 90°) on a flat sea 
surface. For a rough surface, shadowing and 
multiple reflections become important factors 
when the sun is low. The lower and upper 
branches of the curve marked cr = 0.2 represent 
two assumptions regarding the effect of multiple 
reflection. True values are expected to lie 
between the indicated limits [38]. 




Fig. 5-94. The radiance of the sea surface, 
N(/u), divided by the sky radiance at the 
zenith, N*(0), as a function of the vertical 
angle pi. The curves are computed for a flat 
(cr = 0) and rough (cr = 0.2) surface for two 
of the sky conditions illustrated in Fig. 5-93 
[38], 


VERTICAL ANGLE p 


900 






MARINE BACKGROUNDS 169 

Examples of the spectral radiance of the sea for day and after sundown are shown 
in Figs. 5-95, 5-96, and 5-97 [1]. 

For further data on sea-surface geometry see [40]. 



WAVELENGTH (p) 

Fig. 5-95. Spectral radiance of the Banana 
River at Cocoa Beach, Florida [33], 



Fig. 5-96. Spectral radiance of the 
ocean [33]. 


Fig. 5-97. Spectral radiance of the ocean 
versus elevation angle [33]. 



5.9.3. Sea-Surface Temperature Distribution. The temperature of the sea sur¬ 
face determines the contribution of emission to its total radiance. In arctic regions 
this temperature is near 0°C; near the equator it rises to 29°C. Currents such as the 
warm water of the Gulf Stream produce anomalies of several degrees centrigrade as 
it flows into colder areas. However, in most infrared scenes of marine interest, it is 
the radiance variation from point to point that determines the background against 
which a target is seen. Recent improvements in "thermal mappers” have shown details 
of this variation which is usually caused by temperature differences over the sea sur¬ 
face, but under some conditions reflected sky radiance predominates. 

The temperature of the upper 0.1 mm of the sea surface under evaporative conditions 
has been measured as 0.6°C colder than water a few centimeters below [39]. The 












170 


BACKGROUNDS 



TEMPERATURE (°C) 

Fig. 5-98. Thermal structure of the sea boundary layer. Previous conditions: 12 hr 
cool (12° to 15°C), no rain. Data taken during passage of warm front. 

sharpest gradient is in the upper 1 mm [40]. Measurements typical for the condi¬ 
tions noted are shown in Fig. 5-98. 

The temperature of this layer with low heat capacity is determined by the rate of 
evaporation, by radiation exchange, and by the flow of heat from the air and from 
below. It has been found experimentally that the presence of surface contaminations 
reduces (slightly) the flow of heat from below so that a "slick” (a region in the sea with 
enough surface contamination to alter surface tension) appears colder than adjacent 
areas outside the slick. 

Finally, the flow of heat from below is also influenced by the convective activity of 
the water layer above the thermocline. 

5.9.4. Sky Radiance. For examples of sky radiance at night under clear, overcast, 
and other conditions refer to Sec. 5.2. 

NOTES ADDED IN PROOF 

5.1. Sky Backgrounds. There is a moderate amount of literature on the spatial 
and temporal fluctuations of the sky background [41-47]. 

5.2.1. Auroral Spectra. It is difficult to investigate the aurora and airglow beyond 
2.0 lx because of absorption and thermal emission processes in the atmosphere. Ref¬ 
erence [48] gives some predicted values for the 2.0-p, to 3.5-p, region. General reviews 
and one case of an application are covered in [49] through [52]. 






REFERENCES 


171 


5.4.5. Spectral Distribution of Stellar Radiation. A useful source is [53], which 
includes reference [21] as an appendix. Additional work on infrared stellar sources has 
been done at Ohio State [54,55]. Reference [56] is a helpful catalog. Reference [57] 
updates the parameters used in converting from visual magnitudes to infrared spectral 
irradiance. Reference [58] is an example of an application. Other recent work is 
that of Low and Johnson [59] in the 10-/x to 20-p region and of Leighton [60]. Reference 
[61] gives a 5-color statement of the magnitudes of 1300 bright stars. 

One must at present use the visual magnitudes of bright stars to compute their 
infrared irradiance simply because the enormous job of cataloging the infrared emission 
from all the stars has hardly begun. The present approach will not adequately predict 
the irradiance from massive cool stars whose infrared magnitude might far exceed their 
visual magnitude [60]. 

5.5. The Earth as a Background. A comprehensive review of the unclassified 
literature on earth-background measurements taken from aircraft, satellites, rockets, 
and balloons is contained in [62]. 

5.8.2. Terrain Emissivity and Reflectivity. A complete catalog of spectral re¬ 
flectance data of terrain from all available sources, reduced to a standard format of 
presentation, is now available [35]. The report itself is very bulky and has had limited 
circulation. However the data is on file at the Target Signatures Analysis Center, 
Willow Run Laboratories, at The University of Michigan’s Institute of Science and 
Technology. 

References 

1. E. E. Bell, L. Eisner, J. Young, and R. A. Oetjen, J. Opt. Soc. Am., 50, pp. 1313-1320 (Dec. 1960). 

2. E. E. Bell, I. L. Eisner, and R. A. Oetjen, "The Spectral Distribution of the Infrared Radiation 

from the Sky,” Proc. of the Symposium on Infrared Backgrounds, Nonr-1224(12), Engineering 
Research Institute, University of Michigan (March 1956). AD 121010. 

3. Infrared Satellite Backgrounds, Part I: Atmospheric Radiative Processes, AFCRL 1069(1), 
The Boeing Co., Aero-Space Div., Seattle, Wash. (Sept. 30, 1961). 

4. J. W. Chamberlain, Physics of the Aurora and Airglow, (Academic Press, 1961). 

5. A. W. Harrison and A. V. Jones, J. Atmos. Terres. Phys., 11, pp. 192-199 (1957). 

6. A. W. Harrison and A. V. Jones, J. Atmos. Terres. Phys., 13, pp. 291-294 (1957). 

7. E. H. Vestine, Terr. Magn., 49, pp. 77-102 (June 1944). 

8. F. W. G. White and M. Geddes, Terr. Magn., 44, pp. 367-377 (Dec. 1939). 

9. L. Harang, The Aurorae (John Wiley & Sons, Inc., New York, 1951). 

10. C. Stormer, The Polar Aurora (Clarendon Press, Oxford, 1955). 

11. E. L. Krinov, "Spectra Reflectance Properties of Natural Formations.” Originally published 
in Russian, 1947. Translation: National Research Council of Canada. Tech. Trans. TT439, 
Ottawa, Canada (1953). 

12. F. F. Roach, Proc. IRE, 47, p. 267 (1959). 

13. J. F. Noxon, A. Harrison, and A. V. Jones, J. Atmos. Terres. Phys., 16, pp. 246-251 (1959). 

14. A. Vallance Jones and H. Gush, Nature, 172, p. 496 (Sept. 12, 1953). 

15. D. M. Hunten, J. Atmos. Terres. Phys., 7, pp. 141-151 (1955). 

16. M. J. Seaton, J. Atmos. Terres. Phys., 4, pp. 285-294 (1954). 

17. J. C. Brandt and J. W. Chamberlain, J. Atmos. Terres. Phys., 13, pp. 90-98 (Dec. 1958). 

18. J. C. Brandt, Astrophys. J., 128, pp. 118-123 (1958). 

19. J. C. Brandt, Astrophys. J., 130, pp. 228-240 (July 1959). 

20. J. W. Chamberlain and C. Sagan, Planet. Space Sci., 2, pp. 157-164 (1960). 

21. L. Larmore, Infrared Radiation from Celestial Bodies, U.S. Air Force Project, RAND Research 
Memo RM-793-1 (17 March 1952). 

22. Infrared Engineering Handbook, Astrionics Div., Aerojet-General Corp. (25 Aug. 1961). 

23. R. C. Ramsey, "Spectral Irradiance from Stars and Planets above the Atmosphere from 0.1 
to 100.0 Microns,” Appl. Optics, VI, 4, p. 465 (July 1962). 

24. Jen-Hu-Chang, Ground Temperature, Blue Hill Meteorological Observatory (Harvard Uni¬ 
versity Press, Cambridge, Mass., 1958). 

25. F. A. Berry, et al., Handbook of Meteorology (McGraw-Hill Book Co., New York, 1945). 


172 


REFERENCES 


26. I. Solomon, Estimates of Altitudes with Specified Probabilities of Being Above All Clouds, 
Tech. Rept. 159, Air Weather Service (MATS), U.S. Air Force (Oct. 1961). 

27. H. D. Nelgner and J. R. Thompson, Airborne Spectral Radiance Measurements of Terrain and 
Clouds, Emerson Electric Manufacturing Co., St. Louis, Mo., Rept. 1323, April 1962. 

28. R. Schimpf and C. Aschenbrenner, Z. Phot. Wiss. Tech., 2, pp. 41-45 (1940). 

29. Frank Benford, Ilium. Eng. Soc., 42, pp. 527-544 (May 1947). 

30. F. A. Brooks, J. Meteorol., 9, pp. 41-52 (1952). 

31. W. R. Fredrickson, N. Ginsburg, and R. Paulson, Infrared Spectral Emissivity of Terrain, 
Final Report, Syracuse University Research Inst., Syracuse, N.Y. AD 155552. 

32. D. M. Gates and W. Tantraporn, Science, 115, pp. 613-616 (1952). 

33. E. E. Bell, L. Eisner, J. Young, and R. A. Oetjen, J. Opt. Soc. Am., 52, pp. 201-209 (Feb. 1962). 

34. W. R. Fredrickson, H. Ginsburg, R. Paulson, and D. L. Stierwalt, Infrared Spectral Emissivity 
of Terrain, Int. Dev. Rept. No. 2, AF33(616)-5034, Syracuse University Research Inst., Syracuse, 
N.Y. (Aug. 1, 1957). 

35. Target Signatures Study Interim Report, Volume V: Catalog of Spectral Reflectance Data, The 
University of Michigan, Rept. No. 5698-22-T(V), October 1964. 

36. E. D. McAlister, unpublished data, 1951-1952. 

37. H. O. McMahon, J. Opt. Soc. Am., 40, pp. 376-380 (June 1950). 

38. C. Cox and W. Munk, Bull. Scripps Inst. Oceanog. Univ. Calif., 6, pp. 401-488 (1956). 

39. G. C. Ewing and E. D. McAlister, Science, 131, pp. 1374-1376 (May 6, 1960). 

40. E. D. McAlister, J. Opt. Soc. Am., 52, p. 607 (May 1962). 

41. Measurement of Infrared Radiation Gradients in the Sky, Midwest Research Inst., Kansas City, 
Mo., 1953, AD 206 453. 

42. R. C. Jones, Sky Noise—Analysis of Circular Scanning, Polaroid Corp., Cambridge, Mass., 
November 1953. 

43. R. C. Jones, Sky Noise—Its Nature and Analysis, Polaroid Corp., Cambridge, Mass., September 
1953. 

44. R. E. Eisele, Infrared Background Investigation, Rept. AFCRC-TN59-843, Ramo-Wooldridge 
Division of Thompson-Ramo Wooldridge, Inc., Los Angeles, Calif., June 1959. 

45. Infrared Background Investigation, Thompson-Ramo Wooldridge, Inc., Canoga Park, Calif., 
March 1960, AD 236 913. 

46. H. E. Bennett, J. M. Bennett, and M. R. Nagel, The Spatial Distribution of Infrared Radiation 
from the Clear Sky Including Sequences of Sky Maps at Various Elevations, NAVORD Rept. 
6577, U.S. Naval Ordnance Test Station, China Lake, Calif., September 1959. 

47. H. E. Bennett, J. M. Bennett, and M. R. Nagel, Measurements of Infrared and Total Radiance 
of the Clear Winter Sky at Wright-Patterson Air Force Base, Ohio, Wright Air Development 
Center, USAF Air Research and Development Command, Wright-Patterson AFB, Ohio, March 
1957, AD 118 127. 

48. A. V. Jones, Possible Methods for Studying the Auroral Spectrum in the 2.0 to 3.5 Micron Region, 
Saskatchewan University, Saskatoon, November 1959. 

49. D. M. Hunten, "Optics of the Upper Atmosphere,” J. Appl. Optics, 3, 2, February 1964. 

50. Aurorae and Airglow, National Aeronautics Space Administration, Washington, D.C., April 
1964. 

51. I. Sellin, Auroral Radiations in the Infrared, Laboratories for Applied Sciences, University of 
Chicago, October 1961. 

52. R. Chapman, R. Jones, A. Dalgamo, and D. Beining, Investigation of Auroral, Airglow and 
Night Emissions as Related to Space-Based Defense Systems, Geophysics Corp. of America, 
Bedford, Mass., June 1962. 

53. R. G. Walker, Infrared Celestial Backgrounds, Air Force Cambridge Research Labs., Bedford, 
Mass., July 1962. 

54. P. E. Barnhart and W. E. Mitchell, Jr., Stellar Background Measurement Program, The Ohio 
State University, Columbus Research Foundation, Columbus, Ohio, July 1964. 

55. Space Background Study for Project DEFENDER, Eastman Kodak Co., Rochester, N.Y., April 

1963, AD 403 780. 

56. Space Handbook No. 1, ACF Electronics Division ACF Industries Inc., Riverdale, Md., Novem¬ 
ber 1962. 

57. Celestial Background Radiation, Air Force Cambridge Research Labs., Bedford, Mass., March 

1964, AD 602 616. 

58. L. L. Collins and R. B. Freund, Celestial Background Simulation Techniques, Northrop Space 
Labs., Hawthorne, Calif., 1961, AD 282 788. 

59. F. J. Low and H. L. Johnson, "Stellar Photometry at 10 g,” J. Appl. Phys. 139, 1130 (1964). 

60. R. Leighton, Astrophys. J. (Letters), No. 4 (1965). 

61. "Magnitudes and Colors of 1300 Bright Stars,” Sky and Telescope, 30, 1, 24 (July 1965). 


REFERENCES 


173 


62. N. Dittmar, F. Farley, and J. Boyse, Earth Background Measurements: A Survey of the Un¬ 
classified Literature, The University of Michigan, Willow Run Labs., Inst, of Science and 
Technology, Rept. No. 6054-16-X, in preparation. 

63. A. Adel, Observations of Atmospheric Scattering Near the Sun’s Limb, Arizona State College, 
Flagstaff, Ariz., January 1961, AD 273 599. 

64. R. C. Jones and A. M. Nagvi, Satellite Navigation by Terrestrial Occulations of Stars; III: 
Interference due to Brightness of the Earth’s Atmosphere, Geophysics Corp. of America, Boston, 
Mass., October 1962, AD 287 869. 

65. A. M. Nagvi, Satellite Navigation by Terrestrial Occulations of Stars —Considerations Relating 
to Refraction and Extinction, Geophysics Corp. of America, Bedford, Mass., October 1962, 
AD 287 868. 

66. G. Newkirk and J. Eddy, Light Scattering by Particles in the Upper Atmosphere, Colorado 
University, Boulder, May 1963. 

67. R. K. McDonald, "High Altitude Sky Radiance Assessment,” Proc. IRIS, 9, 3, Boeing Co., 
Seattle, Wash, (unclassified article in classified volume). 

68. N. P. Laverty and W. M. Clark, "High-Altitude, Daytime Sky Radiance Measurements,” Proc. 
IRIS, 9, 3, Boeing Co., Seattle, Wash, (unclassified article in classified volume). 


















































































































































Chapter 6 

ATMOSPHERIC PHENOMENA 

Gilbert N. Plass 

Southwest Center for Advanced Studies 

Harold Yates 

Barnes Engineering Company 


CONTENTS 


6.1. Properties of the Atmosphere.177 

6.1.1. Temperature.177 

6.1.2. Pressure.178 

6.1.3. Density.178 

6.1.4. Atmospheric Composition.178 

6.1.5. Particle Concentration and Size Distribution.187 

6.2. Absorption by a Single Line.189 

6.2.1. Single Line with Lorentz Shape.190 

6.2.2. Single Line with Doppler Shape.191 

6.2.3. Single Line with Both Doppler and Lorentz Broadening.192 

6.3. Absorption by Bands.192 

6.3.1. Elsasser Model.192 

6.3.2. Statistical Model.194 

6.3.3. Random Elsasser Model.196 

6.3.4. Quasirandom Model.196 

6.4. Useful Approximations to Band Models.197 

6.4.1. Weak-Line Approximation.197 

6.4.2. Strong-Line Approximation.200 

6.4.3. Nonoverlapping-Line Approximation.201 

6.5. Scattering.202 

6.5.1. Relationship to Field of View.202 

6.5.2. Meteorological Range.203 

6.5.3. Scattering Coefficient.204 

6.5.4. Scattering Coefficient Measurements.206 

6.6. Atmospheric Scintillation.209 

6.6.1. Inhomogeneities in the Atmosphere.210 

6.6.2. Image Boil.212 

6.6.3. Enlargement of the Image.213 

6.6.4. Atmospheric Scintillation Measurements.214 

6.7. Solar Spectrum Measurements.227 


175 


































6.8. Total Absorption (Laboratory Measurements).237 

6.8.1. Total Absorption by C0 2 . 238 

6.8.2. Total Absorption by H 2 0. 244 

6.8.3. Total Absorption by N 2 0. 246 

6.8.4. Total Absorption by CO.249 

6.8.5. Total Absorption by CH 4 .250 

6.9. Infrared Transmission Through the Atmosphere.252 

6.9.1. Horizontal Paths.252 

6.9.2. Slant Paths.261 

6.10. Calculation Procedures.266 


176 












6. Atmospheric Phenomena 


6.1. Properties of the Atmosphere 

6.1.1. Temperature. Standard-atmosphere temperature profiles from 0 to 100 
km and from 0 to 700 km are shown in Fig. 6-1 and 6-2, respectively. The profiles are 
based on the U.S. Standard Atmosphere, 1962 [1]. 



Fig. 6-1. Atmospheric temperature profiles from 0 to 100 km. Based on 
U.S. Standard Atmosphere, 1962 [1]. 


177 



































178 


ATMOSPHERIC PHENOMENA 



Fig. 6-2. Temperature profile from 0 to 700 km. Based on U.S. Standard 
Atmosphere, 1962 [1], 


Supplemental atmospheric temperature profiles [2], which reflect seasonal and 
latitudinal variability up to 90 km, are shown in Fig. 6-3 and 6-4. Winter profiles 
typical of the tropics (15°N), subtropics (30°N), and midlatitudes (45°N) are shown in 
Fig. 6-3; summer profiles for the same areas are shown in Fig. 6-4. The winter and 
summer profiles for 15°N are identical and actually represent a mean annual profile; 
at this latitude, the temperature-height structure remains relatively constant through¬ 
out the year. 

6.1.2. Pressure. Standard-atmosphere pressure from 0 to 100 km is shown in 
Fig. 6-1; the left ordinate of Fig. 6-1 lists pressure versus height from 0 to 100 km. 
All of the pressure data are based on the United States Revised Atmosphere, 1962 [1], 

Supplemental atmospheric-pressure data, which reflect seasonal and latitudinal 
variability (Sec. 6.1.1), are given in [2]. The differences in pressure between the 
supplemental atmospheres, and between each supplemental atmosphere and the 
standard atmosphere (Fig. 6-1), are slight. 

6.1.3. Density. Standard-atmospheric-pressure data, which reflect seasonal and 
latitudinal variability (Sec. 6.1.1), are given in [2] and Fig. 6-1. 

6.1.4. Atmospheric Composition. Table 6-1 gives the composition of the atmos¬ 
phere up to about 90 km. Only carbon dioxide (C0 2 ), water vapor (H 2 0), ozone (0 3 ), 
methane (CH 4 ), nitrous oxide (N 2 0), and carbon monoxide (CO), are discussed in this 
section. The two most abundant gases, N 2 and 0 2 , although they do not exhibit any 
infrared absorption bands, affect the intensities of the observed absorption bands of 
the other constituents through Lorentz (pressure, collision) broadening (Sec. 6.2). 




GEOMETRIC ALTITUDE 


PROPERTIES OF THE ATMOSPHERE 


179 



Fig. 6-3. Supplemental winter temperature profiles for tropics (15°N), 
subtropics (30°N), mid-latitudes (45°N), and subarctic latitudes (60°N) [2]. 


Table 6-1. Composition of the Atmosphere [3]. 


Constituent 

Percent by 
Volume 

Constituent 

Percent by 
Volume 

Nitrogen 

78.088 

Krypton 

1.14 X 10- 4 

Oxygen 

20.949 

Nitrous Oxide 

5 x 10- 5 

Argon 

0.93 

Carbon Monoxide 

20 x lO' 6 

Carbon Dioxide 

0.033 

Xenon 

8.6 x 10- 6 

Neon 

1.8 X 10- 3 

Hydrogen 

5 x 10- 6 

Helium 

5.24 x 10- 4 

Ozone 

variable 

Methane 

1.4 X 10- 4 

Water Vapor 

variable 


GEOMETRIC ALTITUDE 


















180 


ATMOSPHERIC PHENOMENA 



w 

B 

H 

>—c 

H 

< 


O 

2 

H 


O 


w 

o 


Fig. 6-4. Supplemental summer temperature profiles for tropics (15° N), 
subtropics (30°N), mid-latitudes (45°N), and subarctic latitudes (60°N) [2]. 


6.I.4.I. Carbon Dioxide Distribution. The average amount of C0 2 present in the 
atmosphere is 0.33% by volume. This average concentration remains almost constant 
in both space and time (Table 6-2). The burning of fossil fuels gives rise to a gradual 
increase of about 0.7 ppm per year. Superimposed on this fundamental cycle is a 
ground-level daily cycle caused by the exchange of C0 2 with soil and vegetation. The 
ground-level daily cycle can have local fluctuations of C0 2 concentration ranging from 
200 to 600 ppm, but is characteristically confined to a shallow layer of the atmosphere 
immediately above the earth’s surface (up to a few hundred feet). In addition, there is 
a seasonal variation, apparently due to vegetation, which depends on latitude and is 
about 2 ppm between 45° and 90°N. 

In view of both the small departure in C0 2 concentration over the longer cyclic 
period and the uniformity of horizontal C0 2 distribution, it may be assumed that no 
noticeable variations of the mixing ratio occur with height above the biologically active 



















PROPERTIES OF THE ATMOSPHERE 181 

ground layer. Thus, the annual mean C0 2 concentration profile shown in Fig. 6-5 
is reasonably accurate, at least above a few hundred feet altitude. 

Below this altitude, data on the distribution of C0 2 over a particular area at a particu¬ 
lar time for absorption and transmission calculations is desirable. In most cases, 
however, such data are not available and the mean profile will probably have to used, 
with some degradation in accuracy expected. 


Table 6-2. Variation of C0 2 Concentration [41. 


Air Mass 
Time of Year 
Location 


Average Variation 
from Average 
Concentration 

(%) 


Maritime air (Europe) —1.1 

Continental air +1.6 

Polar air —0.7 

Tropical air +2.5 

Spring (Europe) +0.8 

Summer —1.4 


Air Mass 
Time of Year 
Location 

Autumn 

Winter 

Rural France (64° N) 
West Indies (20° N) 
South America (40° S) 
Cape Horn (56° S) 


Average Variation 
from Average 
Concentration 
(%) 

- 0.1 
+0.6 
+4.4 
+3.3 
- 1.0 
—6.9 



6.I.4.2. Water-Vapor Distribution. Water-vapor measurements are generally in 
agreement below the tropopause. In the stratosphere, however, there is wide dis¬ 
agreement among various measurements [5]. Some indicate a relatively dry at¬ 
mosphere with a constant mixing ratio of about 0.05 g/kg (dry stratosphere); other 
measurements indicate that there is a recovery of the mixing ratio from about 0.002 or 
0.003 at the tropopause to about 0.1 near 30 km (wet stratosphere). 

There is no agreement at the present time between those who favor a "dry” strato¬ 
sphere [6-16] and those who favor a "wet” stratosphere [17-24]. Accordingly, water- 
vapor distribution for a dry stratosphere is shown in Fig. 6-6 and for a wet stratosphere 
in Fig. 6-7 and 6-8. 



MIXING RATIO (g precipitable H 9 0 vapor per kg dry air) 


182 


ATMOSPHERIC PHENOMENA 



ALTITUDE (km) 


Fig. 6-6. Water vapor mixing ratio vs. altitude 
("dry” stratosphere) [25]. 


RELATIVE HUMIDITY (%) 







PROPERTIES OF THE ATMOSPHERE 


183 



N - Mean of 2 hydrometric ascents 
by U. S. Naval Research Labs. 

B - Mean of 2 hydrometric ascents 
by Ballistic Research Labs. 

F - One spectroscopic ascent by Univ. 
of Denver 

J - Means of the Japanese Meteorological 
Agency ascents, 100 of which reached 
or exceeded 300 mb and 2 of which 
reached 10 mb 


M - Means of the British Meteorological 
Research Flight hydrometric ascents, 

400 of which reached or exceeded 300 mb 

V - Mean of 7 ascents of the United Kingdom 
Atomic Energy Authority's water vapor 
absorption device (vapor trap) 

X - Mean of all symbols at level 


Fig. 6-7. Water vapor mixing ratio vs. altitude ("wet” stratosphere) [26]. 



184 


ATMOSPHERIC PHENOMENA 



TEMPERATURE (°C) 


N - Mean of 2 hydrometric ascents 
by U. S. Naval Research Labs. 

B - Mean of 2 hydrometric ascents 
by Ballistic Research Labs. 

F - One spectroscopic ascent by Univ. 
of Denver 

D - One hydrometric ascent by Univ. 
of Denver 

J - Means of the Japanese Meteorological 
Agency ascents, 100 of which reached 
or exceeded 300 mb and 2 of which 
reached 10 mb 


M - Means of the British Meteorological 
Research Flight hydrometric ascents, 

400 of which reached or exceeded 300 mb. 

V - Mean of 7 ascents of the United Kingdom 
Atomic Energy Authority's water vapor 
absorption device (vapor trap) 

X - Mean of all symbols at level 


Fig. 6-8. Water vapor dewpoint and frost point vs. altitude ("wet” stratosphere) [26], 


























PROPERTIES OF THE ATMOSPHERE 


185 


6.I.4.3. Ozone Distribution [24]. Of the ozone in the atmosphere 90% is concen¬ 
trated in a layer between about 10 and 30 km above the earth’s surface. The maximum 
concentration occurs between about 25 and 30 km. 

In the upper atmosphere, ozone is formed by the photochemical dissociation of oxygen 
caused by radiation at wavelengths shorter than 2530 A. The oxygen atoms subse¬ 
quently combine with an oxygen molecule to form ozone. In the lower atmosphere, 
minor amounts of ozone are thought to be formed through photochemical reduction of 
atmospheric pollutants. Ultraviolet radiation between 2000 and 2900 A breaks down 
ozone, resulting in an equilibrium in ozone formation and destruction. Should an 
imbalance in ozone concentration at any particular altitude occur, the time required to 
restore equilibrium can be determined. At altitudes of about 50 km, this time is a 
matter of minutes, below 35 km a matter of days, and below 15 km a matter of years. 

Low-altitude ozone concentrations are influenced by atmospheric motions, dust, and 
other attenuators that interfere with the establishment of equilibrium conditions 
[4, 27-29]. For example, a large-scale flattening out causes a low-altitude secondary 
maximum in the ozone profile near the tropopause. When such conditions occur, 
photochemical processes quickly restore equilibrium in the higher altitudes and pro¬ 
vide an overall increase in the total ozone content of the atmosphere. Hence, as shown 
in Fig. 6-9, a correlation exists between the maximum level of ozone and the total 
amount of ozone [30]. Two levels of maximum concentration are indicated: (1) a high- 
altitude (27 km) maximum of about 10 to 15 x 10 3 cm km -1 that occurs during all 
seasons, and (2) a low-altitude maximum (12 km) that occurs only during the winter 
and only at times of high total ozone concentration. 



Fig. 6-9. Mean seasonal vertical distribu¬ 
tion of ozone (approximately 52°N) [29], 


Table 6-3, based on limited data, presents probable ozone distributions. The large 
increase at the 12-24-km altitude reflects the winter correlation between total ozone 
and lower stratospheric temperature changes (sudden warnings). 


Table 6-3. Vertical Distribution 
of Ozone for Two Total Ozone 
Concentrations [31]. 


Total 

Ozone 

(cm) 

0-12 

Altitude (km) 

12-24 24-36 

36-54 

0.300 

15% 

37% 

37% 

11% 

0.500 

12% 

53% 

25% 

8% 






186 


ATMOSPHERIC PHENOMENA 


120 


100 


80 


s 


•o 60)— 

3 


40 


20 




No, 



NOTE: At each 10 km level, max. 

& min. deviations to daytime 
mean model indicated by 
dashes ' 




?s) 


’'s. 


■* x=.200 cm Models for 70°N based on 

- *=.340 cm Seasonal Variation of Total 

- *=.400 cm Ozone Amount 

- Secondary Maximum indicated by 
Venkateswaren Using Echo I 

L 



/) Primary 
Maximum 
/ 

J/ 


10 


,-7 


,-6 


,-5 


.-4 


,-3 


10'‘ 10' u 10"“ 10” 10 “ 10 
Ozone Concentration (cm km *) 

Fig. 6-10. Mean ozone distribution [24]. 


-2 


10 


-1 


Figure 6-10 shows the mean ozone distribution for the Northern Hemisphere. Fig¬ 
ure 6-11 and 6-12 show the seasonal variation of total ozone and the mean meridional 
distributions of ozone, respectively. The curves in Fig. 6-10 are essentially an envelope 
of possible distributions within which it is presumed all distributions will occur. The 
primary maximum concentration shown at an altitude of 25 to 30 km is well founded 
on both theoretical and experimental data. The remainder of the profile, however, 
is established in part by theoretical considerations of the effect of the normal process 
of weather on ozone distribution [27]. 

Figure 6-10 also shows suggested maximum and minimum concentration extremes 
for each 10-km level based on highest and lowest concentration values on record. 
These minimum and maximum extremes should not be interpreted as minimum and 
maximum profiles. 

The nocturnal high-altitude buildup of ozone concentrations shown in Fig. 6-10 
is, to a great extent, conjecture. Theory predicts that such a secondary maximum 
should occur at about 70 km at night, primarily as a result of the large concentrations 
of atomic oxygen at these heights [32, 34]. Some evidence of the existence of such a 



Fig. 6-11. Seasonal variation of total ozone [32], 









PROPERTIES OF THE ATMOSPHERE 


187 



Latitude (°N) 

Fig. 6-12. Mean meridional distribution of ozone [33]. 

high-altitude maximum is presented by spectral observations of Echo I brightness [35]. 
In this instance, the high-altitude maximum occurs at about 55 km, and can be ex¬ 
plained only partially by photochemical theory. System inaccuracies and questionable 
assumptions may account for some of the observed results. 

6.I.4.4. Methane, Nitrous Oxide, and Carbon Monoxide Distributions. Methane, 
nitrous oxide, and carbon monoxide are rarer gases that absorb significantly only over 
long paths. The amount of these gases in the atmosphere is given in Table 6-1. All 
available evidence indicates a uniform mixing ratio [36]. 

6.1.5. Particle Concentration and Size Distribution. The concentration and size 
distribution of scattering particles in the atmosphere vary widely both geographically, 
and temporally in a given location. Accurate measurement of these particles is diffi¬ 
cult, and the amount of good data is limited. From the available data, however, repre¬ 
sentative samples of the distribution of particles with respect to size are plotted in Fig. 
6-13 through 6-16. Figure 6-13 gives an average particle-size distribution curve for a 
continental air condition and for a maritime air condition [37]. Figure 6-14 is the dis¬ 
tribution of particles measured by capture for a haze and a fog [38]. Figure 6-15 is the 
relative size-distribution curve for a fair-weather cumulus cloud, where the total num¬ 
ber of particles per cm 3 in the cloud is 300 [39]. So that measured values may be com¬ 
pared with analytical functions used to approximate the size-distribution curves of 
natural aerosols, Fig. 6-16 shows three such functions, two haze models and one cloud 
model [40]. The curves are normalized so that the integrated area under each curve 
gives 100 particles per cm 3 . The concentration of scattering particles as a function of 
altitude is shown in Fig. 6-17, which includes the results of a number of investigators [41]. 




































RELATIVE CONCENTRATION (cm 


188 


ATMOSPHERIC PHENOMENA 



Fig. 6-13. Particle-size dis¬ 
tribution for continental and 
maritime air [37]. 


z 

o 

HH 

< 



Fig. 6-15. Relative size-distribution 
of droplets in a fair-weather cumulus 
cloud [39]. 



Fig. 6-14. Particle size distri¬ 
bution measured by capture for 
a haze and a fog [38]. 



Fig. 6-16. Three model size-distribution 
functions [40], 


1 

0 

0 . 

0 

0 














ABSORPTION BY A SINGLE LINE 


189 



Fig. 6-17. Relative particle concentration as a 
function of altitude from various investigators 
[41]. 


6.2. Absorption by a Single Line 

The fractional radiant absorptance, A, over a finite wave-number interval, <\v, is 
defined as 


A 



(1 -e-V) dv 


( 6 - 1 ) 


where k v is the absorption coefficient at the wave number and u is the mass of absorbing 
gas per unit area. For most atmospheric absorption problems, the Lorentz, or pressure- 
broadened, line shape should be used. At very great altitudes in the stratosphere, the 
principal cause of line broadening is the Doppler effect. 

The absorption coefficient, k v , for the Lorentz line shape is 


_ S a 

tt {v — i^o ) 2 + a 2 


( 6 - 2 ) 


where S is the total line intensity and a is the half-width of the spectral line whose 
center is located at the wave number v 0 . According to kinetic theory, the half-width 
depends on both the pressure, p, and absolute temperature, T, as 


oc = a 0 



1/2 


(6-3) 


where 0 refers to the value of the quantity for some standard condition. For most 
atmospheric infrared problems, pressure, p, can be taken as the total pressure. It is 
more accurate to use an effective pressure equal to the total pressure plus some con¬ 
stant times the partial pressure [42]. If the absorbing gas is only a small fraction of 
the total, however, the difference between the two quantities is usually small. 







190 


ATMOSPHERIC PHENOMENA 


When line broadening is due to the Doppler effect, the absorption coefficient is [43] 

- ir D exp [ “ s? ( "“ (6 - 4> 


where the Doppler half-width is given by 


= i n 2 ) l?2 (6-5) 

and k is the Boltzmann constant, m is the mass of the molecule, and c is the velocity of 
light. Although the Lorentz line shape decreases only as (y — y 0 ) 2 at wave numbers 
that are more than several half-widths from the line center, the Doppler line shape is 
concentrated more near the line center and falls off exponentially in the wings of the 
line. 

When both the processes that lead to the Lorentz and Doppler line shapes must be 
considered at once, the absorption coefficient is given by [43, 441 



(In 2) 1/2 Sa r e* 2 

7 t 312 A vd J- x a 2 + (w — x) 2 

(6-6) 

where 

a = (In 2) 1/2 

Ay^ 

(6-7) 


w=(\n 2 V' 2 ^-^ 1 
ixvd 

(6-8) 


x = Su/2na 

(6-9) 


Although this integral cannot be evaluated in closed form, numerous approximations 
are given in the literature. A general expression for the Taylor series and the asymp¬ 
totic expression is given in [44]; [43] gives a general review of the problem. 

6.2.1. Single Line with Lorentz Shape. When the pressure-broadened Lorentz 
line shape is valid, the absorptance as obtained from Eq. (6-1 and 6-2) can be written as 


A Ay 


=a L [ 1 -“ p (r£) 


dv' 


( 6 - 10 ) 


where v' = (v— v 0 )la (6-11) 

The limits of integration have been extended to infinity since it is assumed that there 
is no absorption outside of the interval Ay. 

Two limiting results for the absorptance that can frequently be used are the weak- 
line and strong-line approximations (Sec. 6.4.1 and 6.4.2). When the path length is 
small or the pressure is large, the absorption is small at all wave numbers, including 
the line centers. In this case the exponential in Eq. (6-10) can be replaced by the first 
two terms in its series expansion, and the fractional radiant absorption, A, over a finite 
wave-number interval, Ay, can be expressed as [45-47] 

A Ay = 2 7 rax = Su for x < 0.2 (6-12) 

Thus, the absorptance for a single line increases linearly as the path length and the 
line intensity increases whenever the weak-line approximation is valid. Equation 
(6-12) is accurate within 10% when x < 0.2. 













ABSORPTION BY A SINGLE LINE 


191 


When either the path length is large or the pressure is small, the absorption may be 
complete over a wave-number region of several half-widths around the line center. 
In this case, the strong-line approximation is valid, which is equivalent to neglecting 
the factor unity compared to v' 2 in the denominator of the exponential in Eq. (6-10). 
This can be done because the factor unity can always be neglected in the wings of the 
lines when v' > > 1. For other values of v', the exponential has a value very close to 
zero so that its exact value does not matter. When this factor is neglected [45-47], 

A = 2(Sau) l/2 for x > 1.63 (6-13) 

which is called the square root region, since the absorptance varies as the square root 
of the path length, pressure, and line intensity. Equation (6-13) is accurate within 
10% when x > 1.63. 

The integral in Eq. (6-10) can be evaluated exactly to obtain the following expression 
for the absorptance of a single spectral line [48]: 

A \v = 2TT(xxe~ x [/o(x) +/i(x)] (6-14) 

where x is defined by Eq. (6-9) and 7 0 and U are the Bessel functions of imaginary 
argument. This function is tabulated in [45]. The limiting expressions given by 
Eq. (6-12) and (6-13) can be obtained from Eq. (6-14) from the usual expansions of the 
Bessel function for small and large values of the argument. 

6.2.2. Single Line with Doppler Shape. When the spectral lines have the Doppler 
shape, the absorptance is obtained by substituting Eq. (6-4) into Eq. (6-1). When the 
weak-line approximation is valid, the absorptance is given by [49] 

A\v = Su (6-15) 


This is the same as Eq. (6-12); thus, the line shape does not affect the absorption when 
the lines are weak. Only the total line strength is of importance in this limiting case. 
When the strong-line approximation is valid, 


where 


A = 2 A vi 


In Jt/A 1/2 

In 2) 


for xo > > 1 


(6-16) 


Xd = 


In 2 \ 1/2 Su 

7T ) \v D 


(6-17) 


The absorptance increases very slowly as the number of absorbing molecules in¬ 
creases because the Doppler line shape drops off exponentially in the wings of the line. 
When the path is sufficiently long to absorb most of the radiation near the center of the 
line, it is not possible to absorb much additional radiation in the wings of the line by 
increasing the amount of absorbing gas. For long paths, the Doppler line shape acts 
qualitatively as though it absorbed all of the radiation over a bandwidth of 2kv n and 
none outside this interval. By contrast, the absorptance increases as the square root 
of the path length for the Lorentz line shape since the line shape varies as (v - ^o)~ 2 
in the wings. 

When neither of these limiting expressions for the absorptance by a line with the 
Doppler shape is valid, it is necessary to use one of the more general expressions that 
have been derived or one of the many tables or graphs that have been calculated [43]. 





192 


ATMOSPHERIC PHENOMENA 


6.2.3. Single Line with Both Doppler and Lorentz Broadening. When the line 
shape is a result of both Doppler and Lorentz broadening (Eq. 6-6), the absorptance is 
still given by Eq. (6-15) when the weak-line approximation is valid. When the strong¬ 
line approximation is valid, the absorptance is given by [49] 


A kv= 2(Sau) 112 



2 

3 / Sau In 2 


(6-18) 


where a is derived from Eq. (6-12). Higher-order terms are given in [49]. The leading 
term in this expression is the same as that of Eq. (6-13), which is the expression for 
the absorptance of a line with the Lorentz line shape in the square root region. In the 
strong-line region, the Lorentz line shape is much more important in determining the 
absorptance than the Doppler line shape because the Lorentz shape falls off much more 
slowly in the far wings of the line. The Doppler half-width may even be much larger 
than the Lorentz, yet the Lorentz line shape can still determine the absorptance. 

More complicated expressions for the absorptance in intermediate regions are dis¬ 
cussed in [43]. 

6.3. Absorption by Bands 

When the spectral lines in a band do not overlap appreciably, the absorptance of a 
group of spectral lines can be calculated by summing the contribution from the in¬ 
dividual spectral lines. However, in many cases of practical interest, the spectral 
lines overlap appreciably and this effect must be taken into account when the absorp¬ 
tance is calculated. When the lines overlap, the absorptance is always less than would 
be expected from the same number of isolated spectral lines. 

The absorptance of a band of overlapping spectral lines depends on details of the 
relative spacing between the spectral lines and their intensity variation. Because of 
the many rapid variations of the absorption coefficient of a band as a function of fre¬ 
quency, it is very difficult to integrate Eq. (6-1), even with a large electronic computer. 
This is not necessary, however, since four models are available that represent the 
absorption from an actual band with reasonable accuracy. These four models are: 
(1) Elsasser model [50]; (2) statistical model [51, 52]; (3) random Elsasser model [46,53]; 
(4) quasirandom model [54]. 

6.3.1. Elsasser Model. The Elsasser model assumes that the spectral lines are 
evenly spaced and that they all have the same intensity (Fig. 6-18A). Some portions 
of the C0 2 spectrum can be represented with fair accuracy by this model. However, 
there are always numerous weak lines between the stronger, regularly spaced lines 
in this spectrum and furthermore the line intensity varies with frequency. The weak 
lines absorb an increasing share of the radiation as the path length becomes longer. 
Thus the Elsasser model does not represent C0 2 absorption accurately over a wide range 
of path lengths. 

The exact expression for the absorptance of an Elsasser band is [50] 


where 


A = 1 


1 r — (3x sinh (3 
2tt J _ n eX P cosh /3 — cos z 

(3 = 27 raid 


(6-19) 

( 6 - 20 ) 


x is defined by (6-9), and d is the line spacing. This integral cannot be evaluated in 
closed form, but useful expressions for the absorptance can be found in certain limits 
(Sec. 6.4). An extensive calculation of the integral in Eq. (6-19), including a table 





ABSORPTION BY BANDS 


193 



Frequency 


Fig. 6-18. Comparison of Elsasser, statistical and random Elsasser model. 


for values of (3 from 0.0001 to 1.0 and for values of Su/d sinh (3 from 0.02 to 1.5(10) 5 is 
given in [55]. The variation of the absorptance as a function of (3 2 x is shown in Fig. 
6-19. The various curves are for particular values of /3; i.e., they give the absorptance 
at constant pressure. The linear and square root approximations are indicated to show 
their regions of validity. 



Fig. 6-19. Absorption for an Elsasser band as a function of (3 2 x = 2iraSu/d 2 . 















194 


ATMOSPHERIC PHENOMENA 


6.3.2. Statistical Model. The statistical or Mayer-Goody model assumes that the 
spectral lines have a random spacing (Fig. 6-18B) as contrasted to the regular spacing 
of the Elsasser model. The intensity of the spectral lines can vary in any manner what¬ 
soever as long as it can be represented by some distribution function. The absorptance 
of H 2 O can be represented by the statistical model over a moderate range of path lengths 
and pressures. 

The absorptance over a wave-number interval, D, when the statistical model is valid, 
is given by [46] 

4 = 1- (l-A sl , D ) n (6-21) 

where 


A s i,d (S 0 ,^,p) = f A si, D (S, fx,p)P(S,S 0 ) dS 

•'o 

Asi'D is the absorptance of a single isolated spectral line over the wave-number interval 
D, P{S,S 0 ) is the normalized probability of finding a spectral line with an intensity 
between S and S + dS, S 0 is a parametric mean line intensity that occurs in the intensity 
distribution function, and n is the number of spectral lines in the interval D whose mean 
spacing is d. 

When n in the interval D is large, Eq. (6-21) approaches the form [46, 51, 52] 

A — 1 — exp (— uAsI'D) for n » 10 (6-22) 


Equation ( 6 - 22 ) should not be used when there are a small number of lines in the 
interval. 

As examples of the use of these equations assume that: (a) all the spectral lines are 
equally intense, so that 

P(S) — 8(S — So) (6-23) 

( 6 ) an exponential distribution of line intensities, 

P(S) = So” 1 exp (- S/So) (6-24) 


For intensity distribution (a), the absorptance of a single line averaged over the distri¬ 
bution is obtained by replacing S by S 0 in the appropriate expression for the absorptance 
of a single line given (Sec. 6-2). This result can then be substituted into Eq. (6-21) or 
(6-22) to obtain the absorptance of a band. The absorptance for this case is shown in 
Fig. 6-20, assuming that all of the spectral lines are equally intense, that the Lorentz 
line shape is valid, and that the absorption is from a large number of spectral lines 
with an average spacing d. 

The single-line absorptance as calculated for intensity distribution ( b ) is [46] 


4 = 1 — 


fixp 

n{ 1 + 2 x 0 ) 1/2 


(6-25) 


or 

for n » 10 (6-26) 

where 

x 0 = SoAt/27ra 

The expressions for the absorptance of a single line that are used in deriving Eq. (6-25) 
and (6-26) assume an infinite frequency interval. These approximate expressions for 
Asi'D may be used only so long as there is no appreciable absorption by the single line 
outside of the interval D. In general, more complicated expressions for the absorptance 


A = 1 — exp 


fix 0 


(1 + 2 x 0 ) 1/2 








ABSORPTION BY BANDS 


195 



Fig. 6-20. Absorption for the statistical model as a function of /3 2 x. 


of a single line over a finite frequency interval must be used [54], The single-line 
expressions for the absorptance of a line with the Doppler shape or the Doppler-Lorentz 
shape can also be substituted into Eq. (6-21) or (6-22). 

The absorptance for the statistical and Elsasser models is compared in Fig. 6-21. The 
absorptance is always larger at long path lengths for the Elsasser model since the 
spectral lines absorb most efficiently with regular spacing. With a random spacing, the 
lines overlap more strongly because lines which happen to be close together cannot 
absorb as efficiently as when they are spaced further apart. 



Fig. 6-21. Comparison of absorption from the statistical and Elsasser band models. 







196 


ATMOSPHERIC PHENOMENA 


The absorptance for the Lorentz, Doppler, and square (absorption coefficient constant 
over a finite frequency interval) line shape are compared in Fig. 6-22 for a random 
distribution of lines. In each case, the statistical model with all of the lines equally 
intense is assumed. For comparison purposes, (3d and xd are set equal to (3 and x for 
the Lorentz line shape. The very slow increase of the absorptance for large path 
lengths can be seen for the Doppler line shape. 



^ D 2 x D =7r 1/2 (Jn 2) 1/2 A^Sud* 2 


Fig. 6-22. Absorption of spectral lines with Doppler, Lorentz, and square line shape. 
Statistical model with equally intense lines assumed. 


6.3.3. Random Elsasser Model. A more accurate representation of band ab¬ 
sorption is provided in many cases by the random Elsasser model, which assumes the 
random superposition of several different Elsasser bands. Each of these superposed 
bands may have a different line intensity and spacing. As many different Elsasser 
bands as desired may be superimposed in this model. Thus, all of the weak spectral 
lines that contribute to the absorption for the path lengths and pressures considered 
can be included in the absorption calculations. The superposition of three different 
Elsasser bands is illustrated in Fig. 6-18C. 

The absorptance of M randomly superposed Elsasser bands is [46, 53] 

M 

A = A Eti (xi, (3i )] (6-27) 

i=l 

where Aea is the absorptance of an Elsasser band with a half-width of a,, a line spacing 
di, and a line intensity Si, so that Xi = S,u/27ra; and /3, = 27ra,7d,. 

6.3.4. Quasirandom Model. The quasirandom model is the most accurate and, 
necessarily, the most complicated of the band models. It is especially useful when the 
absorptance is required over a wide range of path lengths and pressures. The spectral 







USEFUL APPROXIMATIONS TO BAND MODELS 


197 


lines in an actual band are arranged neither as regularly as required by the Elsasser 
band nor in as random a fashion as in the statistical model; there is some order in their 
arrangement. In the quasirandom model the absorptance is calculated first for a fre¬ 
quency interval that is much smaller than the interval size of interest. This localizes 
the stronger lines to a narrow interval around their actual positions and prevents the 
introduction of spurious overlapping effects. The absorptance of this narrow interval 
is calculated from the equation for the single-line absorptance over a finite interval 
[54]. The absorptance for each of the /V spectral lines in the interval is calculated 
separately and the results combined by assuming a random placing of the spectral lines 
within the small interval. The absorption from the wings of lines in neighboring 
intervals is included in the calculation. The results are averaged for at least two 
different arrangements of the mesh that divides the spectrum into frequency intervals. 
Finally the absorptance values for all of the small intervals that fill the larger interval 
of interest are averaged to obtain the final value for the absorptance. An electronic 
computer is commonly used to calculate results for this model when many spectral lines 
are involved. The many weak spectral lines and their relative spacing are accurately 
taken into account by this model. 

The absorptance for the quasirandom model is given by [54] 



where Aj is the absorptance of each of the L smaller wave-number intervals into which 
the original interval Ai> is subdivided. The absorptance Aj is calculated from 

Aj = 1 - n [! - 'U D {i) (Si, di ) ] (6-29) 

i= 1 

where A s i,D (i) is the single-line absorptance over the finite interval D for a line with 
intensity Si and half-width at, and M is the number of lines in the frequency interval j. 

References [56] and [57] give absorptance tables for H 2 0 and C0 2 based on the 
quasirandom model calculations. In these calculations, the lines in the small fre¬ 
quency interval were grouped by intensity decades. The average intensity and number 
of lines in each of these decades were calculated and used in Eq. (6-29). All lines from 
all isotopic species having an intensity greater than 10' 8 of the strongest line in an 
absorption region were included in the calculation. The final results for the trans¬ 
missivity are given every 5 cm -1 for C0 2 from 500 to 10,000 cm 1 and for H 2 0 from 
1000 to 10,000 cm -1 . Values averaged over 20, 50, and 100 cm 1 intervals are also 
presented. All values are given at three temperatures, 200°, 250°, and 300° K, and for 
seven pressures, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1 atm. The path length for CO> 
ranges from 0.2 to 10,000 atm cm and for H 2 0 from 0.001 to 50 cm of precipitable water. 

6.4. Useful Approximations to Band Models 

There are three different limits in which approximate forms of the equations given 
in Sec. 6.2 and 6.3 can be obtained. These are: (1) weak-line approximation; (2) strong¬ 
line approximation; (3) nonoverlapping-line approximation. Each of the approxima¬ 
tions is valid over a wide range of path lengths and pressures. However, care must 
be taken to use each approximation only within its own region of validity. A detailed 
study of these approximations is given in [47]. 

6.4.1. Weak-Line Approximation. The weak-line approximation assumes that 
the absorption due to each line considered individually is small even at the line center. 


198 


ATMOSPHERIC PHENOMENA 


However, the total absorptance due to a number of such lines may have any value 
(even near unity), since the sum of a number of small quantities is not necessarily small. 

For the Elsasser band, Eq. (6-19) reduces to [47] 

A = 1 - e-* 1 (6-30) 

in the weak-line limit. When fix « 1, Eq. (6-30) reduces to Eq. (6-12). When this 
inequality is satisfied, the lines do not overlap and the absorptance increases linearly 
with path length. When the lines begin to overlap, the exponential term in Eq. (6-13) 
takes account of this effect. On a logarithmic plot of absorptance (Fig. 6-19), the slope 
of the absorptance curve is unity in the linear region where the lines do not overlap and 
the weak-line approximation is valid. When this approximation continues to be valid 
as the path length increases, the slope of the absorptance curve decreases, as shown 
on the curves marked (3 = 1 or 10, as the lines begin to overlap. 

The regions of validity of this approximation are shown in Fig. 6-23 and Table 6-4. 
Within the indicated regions, the approximations are accurate within 10%. 


100 



0.001 


OlOOOIL 


0.001 


100 


X = 


Su 

2va 


Fig. 6-23. Regions of validity for band model approximations (Elsasser 
model). 





























































USEFUL APPROXIMATIONS TO BAND MODELS 


199 


Table 6-4. Regions of Validity of 
Various Approximations for Band Absorption.“ 


Approximation 

/3 = 2 Tra/d 


Elsasser 

Model 

Statistical 
Model; All 
Lines Equally 
Intense 

Statistical 
Model; 
Exponential 
Line Intensity 
Distribution 

Strong-line 

0.001 

X 

> 

1.63 

X 

> 1.63 

xq > 2.4 

approximation 

0.01 

X 

> 

1.63 

X 

> 1.63 

xo > 2.4 


0.1 

X 

> 

1.63 

X 

> 1.63 

xo > 2.3 


1 

X 

> 

1.35 

X 

> 1.1 

xo > 1.4 


10 

X 

> 

0.24 

X 

> 0.24 

Xo > 0.27 


100 

X 

> 

0.024 

X 

> 0.024 

x 0 > 0.24 

Weak-line 

0.001 

X 

< 

0.20 

X 

< 0.20 

xo < 0.10 

approximation 

0.01 

X 

< 

0.20 

X 

< 0.20 

xo < 0.10 


0.1 

X 

< 

0.20 

X 

< 0.20 

xo < 0.10 


1 

X 

< 

qo 

X 

< 0.23 

Xo < 0.11 


10 

X 

< 

00 

X 

< 00 

Xo < 00 


100 

X 

< 

00 

X 

< 00 

Xo < 00 

Nonoverlapping 

0.001 

X 

< 

600 000 

X 

< 63 000 

Xo < 80 000 

line 

0.01 

X 

< 

6000 

X 

< 630 

xo < 800 

approximation 

0.1 

X 

< 

60 

X 

< 6.3 

xo < 8 


1 

X 

< 

0.7 

X 

< 0.22 

xo < 0.23 


10 

X 

< 

0.02 

X 

< 0.020 

xo < 0.020 


100 

X 

< 

0.002 

X 

< 0.0020 

xo < 0.0020 


“When x = Su/2ira satisfies the given inequalities, the indicated approximation for the absorption is 
valid with an error of less than 10%. For the exponential line-intensity distribution. x 0 = S 0 ul2na, 
where P{S) =S 0 _1 x exp (—S/S«). 


The weak-line approximation for the statistical model is [47] 


A = 1 — 



(6-31) 


or 

A = 1 - e~** (6-32) 

These equations follow for any intensity distribution. For example, when P(S) is 
given by Eq. (6-24), replace x by x 0 in equations 6-31 and 6-32. The regions of validity 
of this approximation are given in Table 6-4 and are shown in Fig. 6-24. Within the 
indicated regions, the approximations are accurate within 10%. 

The weak-line approximation for the random Elsasser model is [47] 


M 

a =i - n e,z ‘ 

i= l 


(6-33) 


and for the quasirandom model is [54] 


1 L f M a t 

' 4 =fs ‘-no-'") 

j=l L i= l 


(6-34) 




200 


ATMOSPHERIC PHENOMENA 



0.001 0.01 0.1 1.0 10 100 


X = 


Su 

277 0 ? 


Fig. 6-24. Regions of validity for band model approximations (statistical model). 


6.4.2. Strong-Line Approximation. The strong-line approximation assumes that 
the absorption is virtually complete over a bandwidth of at least several half-widths 
around the line center. 

For the Elsasser model, the strong-line approximation is [45] 


A = 4) 



where </> is the error function integral defined as 


(6-35) 


</>U) 



e~ z2 dz 


(6-36) 



































































USEFUL APPROXIMATIONS TO BAND MODELS 


201 


When fi 2 x « 1, Eq. (6-35) reduces to Eq. 6-13, which is the square root approximation 
(Fig. 6-19). For example, the curve marked /3 = 0.01 has a slope of nearly one-half over 
most of the region shown here. Beyond about (3 2 x = 0.05, however, the absorptance 
curve begins to depart from the square root approximation as the spectral lines begin 
to overlap. 

The rather severe limitations of Eq. (6-35) should always be considered. The band 
under consideration must have regularly spaced and equally intense lines. Further¬ 
more, the equation is only valid in the strong-line region, as indicated in Fig. 6-21 and 
Table 6-4. 

The strong-line approximation for the absorptance for the statistical model is [47] 

—M'ffT 


or 


A = 1 — exp — (2/3 2 x/7t) 1/2 for N >> 10 


(6-38) 


where an appropriately defined average value of the line intensity S is used in the 
determination of the value of x. Otherwise, the term in the square bracket of Eq. (6-37) 
may be evaluated for each of the N spectral lines and the results multiplied together. 
The regions of validity of this approximation are given in Fig. 6-24 and Table 6-4. 
For the random Elsasser model, the strong-line approximation is [47] 


A = 




where x t and /3, are as defined after Eq. (6-27). 

For the quasirandom model, the strong-line approximation is [46, 54] 


1 ^ 


j =i 


M 

n 

i =1 


n ll2 zll - <t>(zi)] 


(6-39) 


(6-40) 


where 


Zi 2 = 8cti 2 Xi/D 2 

The expression for the absorptance over a finite frequency interval D [46] has been 
used in deriving Eq. (6-40). 

6.4.3. Nonoverlapping-Line Approximation. The regions of validity for the 
strong- and weak-line approximations are determined by whether the absorption is 
large or small at the frequency of the line centers and do not depend on the degree of 
overlapping of the spectral lines. On the other hand, the only requirement for the 
validity of the nonoverlapping-line approximation is that the spectral lines do not 
overlap appreciably. It is valid regardless of the value of the absorption at the line 
centers. Since there is no effect from the overlapping of the spectral lines, the absorp¬ 
tion is independent of whether the spacing between the lines is regular, random, or 
quasirandom. However, the absorptance can depend on the distribution of line in¬ 
tensities within the band. The nonoverlapping line approximation is particularly 
useful for extrapolating the absorption to small values of path length and of the pres¬ 


sure. 







202 


ATMOSPHERIC PHENOMENA 


For any model, the absorptance when the nonoverlapping approximation is valid 
is [47] 


N 


A \v = ^ 2ira,Xie '[/ 0 (*i) + /i(x,-)] 


i=1 


(6-41) 


where the summation is over the N spectral lines in the frequency interval Aia The 
regions of validity of this approximation are shown in Fig. 6-23 and 6-24 and are given 
in Table 6-4. 

A particularly simple result is obtained if the probability distribution of line in¬ 
tensities is given by Eq. (6-24). Then the absorptance is [47] 

A = 27raxo(1 + 2*o)~ 1/2 (6-42) 


where 


* 0 = So/a/2770 


6.5. Scattering 

Pure scattering occurs if there is no absorption of the radiation in the process, and, 
hence, no loss of energy but only a redistribution of it. Most of the scattering en¬ 
countered in the atmosphere is essentially pure. Typical exceptions to this are the 
dense black smoke issuing from a boiler in which there is incomplete combustion or 
from a volcano emitting large amounts of fly ash. In these cases, the carbon and 
mineral particles not only scatter but also absorb strongly. Only pure scattering is 
discussed in this section. Concepts developed for the visible region of the spectrum 
are used in extending the study of scattering into the infrared region. 

6.5.1. Relationship to Field of View. The attenuation due to scattering of a 
collimated beam of light depends upon the field of view of the receiving instrument. 
If the field of view is very large, some light scattered at a very small forward angle 
will still be accepted and recorded; if the field of view is very small, virtually all scat¬ 
tered radiation can be rejected and only transmitted radiation registered. 

The influence of the field of view of a measuring radiometer or telephotometer is 
discussed theoretically in [58]. Experimental investigations are described in [59] 
and [60]. The dependence of measured results on the field of view appears to be 
adequately described by the empirical relationship [60] 


T e = T+ 0.5 (l-r)(l-e-*) 

where 6 — angular diameter of radiometer field of view (radians) 

To = transmittance measured at a particular X with a radiometer having a field 
of view 6 

T = transmittance of unscattered light 

This relationship agrees well with measurements when the meteorological range 
(Sec. 6.5.2) is greater than 10 km but deviates from measurements in a hazier atmos¬ 
phere. Reference [61] presents tables based on the angular scattering function of a 
normal atmosphere [62], from which the scattered component can be evaluated for a 
given state of the atmosphere. A comparison of predicted values with measured 
values is shown in Fig. 6-25. 


SCATTERING 


203 



Field of View (Degrees) 

Fig. 6-25. Diffuse-to-collimated transmission ratio, 
T<i/T c , as a function of field of view {ad = optical density 
of transmission path) [56]. 


6.5.2. Meteorological Range. Meterological range is a convenient parameter for 
visually describing a particular scattering condition. It is defined as the distance at 
which the average eye can just barely detect a large, black (nonreflecting and nonemis- 
sive) target against the horizon sky. To obtain a definition independent of individual 
eyes, meteorological range is further defined as a 2% contrast between the distant black 
target and the sky. The 2% figure represents a good average for the human eye’s 
capabilities. The meteorological range, on the basis of the 2% contrast figure, is 
related to the visible scattering (Sec. 6.5.3) coefficient, a (km _1 ) by the relationship: 


V = 


\(±\ = 


1.0 


0 .02/ \an 


3.9/0- 


(6-43) 


where V — meteorological range (km) 
a = scattering coefficient (km -1 ) 

The scattering coefficient is related to the transmission (or transmissivity) of a given 
optical path by the relationship: 


T = e~ ax (6-44) 

where T = transmission of optical path of length, x (dimensionless) 
x = optical path length (km) 
a = scattering coefficient (km -1 ) 

Figure 6-26 shows meteorological range versus scattering coefficient. Consider a day 
on which the meteorological range is 20 km. Refer to Fig. 6-26; the visible scattering 
coefficient for this day is 0.0195/km. The transmission over a 5-km path is then: 

T = e -° 195x5 = 0.377 = 37.7% 

A detailed discussion and methods of estimating meteorological range are given in 
Chapters 6 through 9 of [58]. 





204 


ATMOSPHERIC PHENOMENA 



Fig. 6-26. Visible scattering coefficient, G, as a function of meteorological range. 
Solid points are visual ranges taken from the international visibility code. 


6.5.3. Scattering Coefficient. Scattering can be treated theoretically in three 
separate approaches according to the relationship between the wavelength of the 
radiation being scattered and the size of the particles causing the scatter. These 
approaches are: (1) Rayleigh scattering; (2) Mie scattering, and (3) nonselective scat¬ 
tering. 

6.5.3.I. Rayleigh Scattering. Rayleigh scattering applies when the radiation 
wavelength is much larger than the particle size. The volume scattering coefficient 
for Rayleigh scattering can be expressed as [63] 

o- = (4tt 2 NV 2 I\ 4 ) (n 2 - no 2 ) 2 /(n 2 + 2 n 0 2 ) 2 (6-45) 

where N = number of particles per unit volume (cm -3 ) 

V = volume of scattering particle (cm 3 ) 

A. = wavelength of radiation (cm) 

n n = refractive index of medium in which particles are suspended 
n = refractive index of scattering particles 





SCATTERING 


205 


For spherical water droplets in air (n. 0 = 1; n = 1.33 for the visible and near infrared, 
except in the vicinity of absorption bands where anomalous dispersion is encountered), 
Eq. (6-45) becomes 


o- = 0.827A^ 3 /A 4 (6-46) 

where A = the cross-sectional area of the scattering droplet. This expression must be 
integrated over the range of k and A encountered in any given circumstance. As long 
as the original requirement is met for all k and A; i.e., the particle diameter (2 k/AI tt) 
is very small compared to k, the same scattering can be experienced from a large number 
of small particles or a small number of large particles, provided the product NA 3 is the 
same. 

6.5.3.2. Mie Scattering. Mie scattering is applicable where the particle size is 
comparable to the radiation wavelength. The Mie scattering area coefficient is defined 
as the ratio of the area of the incident wave front that is affected by the particle to the 
cross-sectional area of the particle itself. The form of relationship between scattering- 
area coefficient and particle-size parameter is shown in Fig. 6-27. The value of K rises 
from 0 to nearly 4 and asymptotically approaches the value 2 for large droplets. The 
scattering coefficient, a, is related to K by 


a = NKtto 2 


or, for the almost universal condition in which there is a continuous size distribution in 
the particles, by 


cr \ — 7T 


r N(a)K( 

J rt. 


a, n)a 2 da 


(6-47) 



Fig. 6-27. Scattering area coefficient vs. size parameter for 
spherical water droplets (radius = a). Index of refraction, 
n = 1.33 [67]. 




206 


ATMOSPHERIC PHENOMENA 


where cr\ = scattering coefficient for wavelength 

N(a) = number of particles per cubic centimeter in the interval da 
K(a,n) = scattering area coefficient 
a = radius of spherical particle 
n = index of refraction of particle 

With the units of a and N in cm and cm -3 , respectively, a is in units of cm -1 . To 
convert to km -1 , the more commonly used units for a, multiply by 10 5 . 

References [64] and [65] present a detailed treatment of scattering theory. Ref¬ 
erence [66] discusses application of theory for a wide variety of particle composition, 
size, and shape. Tables of the Mie scattering area coefficient are given in [67-69]. 

6.5.3.3. Nonselective Scattering. Nonselective scattering occurs when the particle 
size is very much larger than the radiation wavelength. 

Large-particle scattering is composed of contributions from three processes involved 
in the interaction of the electromagnetic radiation with the scattering particle: (1) 
reflection from the surface of the particle with no penetration; (2) passage through the 
particle with and without internal reflections; and (3) diffraction at the edge of the 
particle. References [70], [71], and [72] discuss the combined effect of all three 
processes, including the interference encountered between the three components, and 
show that, for particles larger than about 2 times the wavelength of the radiation 
(a > 20), the scattering-area coefficient becomes 2, which is the asymptotic value ap¬ 
proached by the Mie coefficient. Thus, the theoretical approach through diffraction, 
refraction, and reflection appears to have little contribution to the more general ap¬ 
proach of Mie. For a < 20, the Mie theory is valid, and for a > 20 the two predictions 
converge on the value 2. A generalized treatment of scattering considered as the sum 
of the refracted, diffracted, and reflected components is given in [58] and [73]. 

6.5.4. Scattering Coefficient Measurements. Reliable scattering-coefficient data 
in the infrared are difficult to obtain because of the contributions of both scattering and 
selective absorption to the extinction coefficient, which is the measured experimental 
value. The procedure usually followed is to confine the measurements to wavelengths 
that are as free as possible from absorption, i.e., in the clearest part of the atmospheric 
windows between the major infrared absorption bands, and assume that the extinction 
coefficient measured is the scattering coefficient. As wavelength increases, however, 
it becomes virtually impossible to find regions completely free of absorption. A further 
assumption, which is valid for water droplets, is that absorption by the scattering 
droplets in the scattering process is negligibly small. 

Figures 6-28 through 6-30 show scattering coefficient as a function of wavelength 
measured over Chesapeake Bay [74]. These curves show a persistent tendency to 
flatten out past 2 /x, which cannot be explained on the basis of any hypothetical particle- 
size distribution. It appears that, past 2 /x, an appreciable amount of absorption by the 
wings of the atmospheric absorption bands is present along with the scattering. There 
is a very slight tendency for the curves obtained on days of high absolute humidity to 
flatten out more rapidly than those obtained on days of low humidity, which indicates 
that the principal absorber is probably water vapor. 

Figure 6-31 is the scattering coefficient measured over a 27.7-km path, essentially 
horizontal, at 10,000-ft elevation between two mountain peaks on the Island of Hawaii 
[74]. The atmosphere was usually very clear, and meteorological ranges of about 200 
mi were common. The tendency to flatten out past 2 /x is again evident. For com¬ 
parison, the values measured on the Nevada desert, another clear atmosphere, are 
included [75]. 


SCATTERING 


207 



WAVELENGTH ( 4 ) 


Fig. 6-28. Scattering coefficient measured over a 5.5-km sea-level maritime path [74]. 


Figures 6-32 and 6-33 show the results of additional scattering measurements over 
Chesapeake Bay [76]. Figure 6-32 contains curves of eight separate measurements 
made on days when the meteorological range extended from 100 km down to 3 km. 
The spread in points for the clearer days is attributable to severe scintillation (twinkle), 
which is normally encountered in a very clear atmosphere. Twinkle can also occur in 
a moist atmosphere but is usually masked. Figure 6-33 gives data for various meteoro¬ 
logical ranges. 

Figure 6-34 gives the scattering coefficients of a haze, and Fig. 6-35 gives the scat¬ 
tering coefficient and optical density of several types of fogs [38]. Optical density, D, 
and scattering coefficient, cr, are related by a = 2.3 D. The data are based on more than 
600 spectrophotometric curves of hazes (optical density per km < 2), small drop fogs 
(optical density per km < 10), evolving fogs (changing size-distribution characteristics), 
and selective fogs and artificial smokes. The data indicate that stable fogs, with a 
density/km between 2 and 20, exhibit essentially the same opacity at all wavelengths 
out to 10 /x, at which point they invariably become sensibly more transparent. Evolving 
fogs and selective fogs, in which the visible density/km again could range from 2 to 20, 
showed a distinctly declining density as the wavelength increased. 

The variation of scattering coefficient (visible) with height up to 10 km is shown in 
Fig. 6-36. 





208 


ATMOSPHERIC PHENOMENA 



Fig. 6-29. Scattering coefficient measured over a 16.25-km sea-level maritime path. 
Curve B represents a typical winter day condition [74]. 


Because the scattering properties of the atmosphere can vary appreciably, it is not 
possible to state a scattering coefficient that will permit accurate predictions over a 
wide variety of conditions. However, a relationship frequently used is: 


o- = c\~y (6-48) 

where c and y are constants determined by the concentration and size-distribution 
values for the aerosol, and A is the wavelength of the radiations. If y = 4, this relation¬ 
ship is recognized as the form of the Rayleigh coefficient. Actually, from the Mie 
theory, y — 4 for very small particles, y = 2 for a particle diameter equal to A; y = 1 for a 
particle diameter equal to (3/2)A, and y = 0 when the diameter is equal to 2A. 

Considering the presence of the atmospheric gases in the aerosol, a more exact form 
of this relationship is [77] 


o- = ak-y + c 2 A- 4 (6-49) 

where the second term accounts for the scattering by the Rayleigh components. In 
most cases, however, the second term is considerably less than the first, and it may be 
neglected for practical purposes. 




ATMOSPHERIC SCINTILLATION 


209 


1 




H 

Z 

H 

o 

i—i 

u< 

6m 

W 

O 

o 

o 


K 

w 

E- 

H 

< 

O 

cn 


1.0 


0.1 


.01 


\ Precipitable Water Vapor in Path (cm) 

^ A - 6 cm 

^ B - 22.7 cm 

\ C - 6.9 cm 

\ 

\ 

\ 

\ 

\ 

\ 



\ 

\ 

\ 

\ 

\ 

\ 

j -1-1-1—\_i—i—i—i— I _i_i_i_ ■ i i i i 


0.1 


1.0 

WAVELENGTH (p) 


10 


Fig. 6-30. Scattering coefficient measured over a 16.25-km sea-level, maritime path. 
Curve B represents a typical summer day condition [74]. 


In typical real atmospheres, the value of y most recommended appears to be very 
nearly 1.3 [78]. It has been suggested [79] that the exponent is related to the mete¬ 
orological range by the empirical form: 

y = 0.0585 {MR) 1 ' 3 (6-50) 

This relationship, and various experimental data [80, 81] are shown in Fig. 6-37. 
The slopes of the data curves from Fig. 6-28 through 6-33 are also shown to give an 
indication of the slope in the infrared. The points are widely spread and significantly 
below the widely accepted value of 1.3. The smaller slopes indicated by the data are 
undoubtedly another manifestation of the contamination of the infrared windows by 
selective absorption. 

6.6. Atmospheric Scintillation [82] 

Scintillation (boil, shimmer, twinkle) is the variation in intensity or angular distribu¬ 
tion of a collimated beam of radiation as a result of inhomogeneities in the atmospheric 
path through which the radiation passes. The following paragraphs briefly discuss 
these variations and their effects on optical systems. Detailed discussions of scintilla¬ 
tion theory are given in [58, 59, 82-88]. 






210 

1.0 


ATMOSPHERIC PHENOMENA 



I 


S 


b 


H 

w 

►—4 

O 

>—i 

Uh 

Uh 

W 


o 


o 

o 

►—I 

CJ 


W 

H 

H 

< 

O 


w 



.01 


0.1 


\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 



1.0 10 

WAVELENGTH (p) 


- Values Obtained at 3 km Elevation in Hawaii 

- Values Obtained at ~1.2 km in Nevada 

Precipitable Water Vapor in Path 
o 11 cm 
• 20 cm 


Fig. 6-31. Scattering coefficient measured over a 27.7-km horizontal path at 3-km 
elevation on the island of Hawaii [74] and the Nevada desert at ground level [75]. 


6.6.1. Inhomogeneities in the Atmosphere. Consider a three-dimensional pattern 
of refractive index inhomogeneities in the atmosphere in which there are discrete 
pattern elements that are somewhat spherical. Also, consider more or less continuous 
sheets of air differing in refractive index from the average, and having surface ir¬ 
regularities. Some of these surface irregularities will be somewhat hemispherical 
and can be treated as discrete elements also. In some cases, the interface on adjacent 
sides of a sheet will be plane, but not parallel. 

Although the interface of these refractive elements may or may not be somewhat dif¬ 
fuse, the transit time for a collimated beam through a refractive element of lower than 
average refractive index will be shortened, while the transit time for the a beam passing 
through a refractive element of higher than average refractive index will be increased. 
Thus, there is a grouping of the "transit times” for the beams having passed through a 
small cubical volume containing a single refractive element. The nature or distribu¬ 
tion of this grouping of transit times is comparable to the distribution of transit times 
for beams that have passed through a similar cubical volume containing an optical 




SCATTERING COEFFICIENT, cr (km 


ATMOSPHERIC SCINTILLATION 


211 




WAVELENGTH ( 4 ) 


Fig. 6-32. Measured values of scattering coefficient Fig. 6-33. Measured values of scatter- 
as a function of wavelength at different times [76]. ing coefficient grouped according to me- 

terological range [76]. 


c 

to 


a> 

■ m 

o 


QJ 

Z 

O 


h£> 

c 

<L> 


Ol 

o 

co 



Fig. 6-34. Scattering coeffi¬ 
cient for a haze [38]. 



C 

<v 


<D 

O 

O 

bT 

C 

• H 

S-* 


aj 

o 

co 









212 


ATMOSPHERIC PHENOMENA 



10~ 3 io" 2 10' 1 10° 

SCATTERING COEFFICIENT, a (km' 1 ) 


Fig. 6-36. Scattering coefficient as a function of height for a variety of conditions in 
clean air. 


element such as a lens or prism. Therefore, the shapes of the wave fronts and the 
angular distributions of the beams are also comparable. To the extent that the re¬ 
fractive elements are in reality comparable to an optical element, they may be so 
treated analytically. 

A single refractive element in the size range of 1 inch to 1 foot in diameter can cause 
an appreciable change in the angular distribution of radiation in a portion of the beam, 
resulting in a marked change in the radiation level at a more distant point in the beam. 
Further, the extent of the change in the angular distribution of the radiation in the 
beam is dependent upon the size, shape, and refractive index differential of the refractive 
element. The extent of change in the radiation level at a more distant point in the 
beam is also dependent, however, on the relative location of the refractive element in the 
optical path, as well as on the total path length. And in some cases where diffraction 
is predominant, the change in the received radiation level is also a function of wave¬ 
length. 

6.6.2. Image Boil. Consider an optical system that is aimed at a small, distant 
source. A region of refractive index inhomogeneities causes nonuniformities of the 
angular orientation of rays within a beam of radiation, and these angular nonuniformi¬ 
ties lead to intensity variations in the plane of the receiver aperture. It follows that 
these nonuniformities in the angular orientation of rays can also produce deviations in 
the distribution of radiation in the image plane of the optical system. As a result, 
the very small image produced under clear, homogeneous conditions can become 
spread out or shifted in position when the incident angular distribution of rays becomes 
spread out or shifted in direction. This effect has been observed photographically and 
visually, and is often referred to in the literature as "image boil.” 

If the rays incident on the optical system are all deviated a like amount and in the 
same direction, as though having passed through a prism, the image of the source 








ATMOSPHERIC SCINTILLATION 


213 



Fig. 6-37. Experimental values of slope of scattering coefficient curves. 


retains its small size but is displaced from the optical axis. Displacements of stellar 
images of as much as 5 to 10 seconds of arc have been measured. Displacements of as 
much as 15 to 30 seconds of arc can be expected under severe ground-to-ground condi¬ 
tions. Because of the erratic motion associated with the observed displacement of the 
image, the term "image jitter” is often used for this effect. 

6.6.3. Enlargement of the Image. To the extent that the rays incident on an 
optical system are either dispersed or converged symmetrically about the direction of 
the optical axis, the image of the small, distant source becomes enlarged. The same 
effect would result if a lens were placed on the optical axis. If the lens were positive, 
the beam would become more convergent and form a small image in front of the focal 






214 


ATMOSPHERIC PHENOMENA 


plane. If the lens were negative, the beam would become more divergent and form a 
small image behind the focal plane. In both cases, the image in the focal plane would 
be spread. The amount of "image spread” is dependent on the broadening of the 
angular distribution of rays that are incident on the aperture of the optical system. 
This broadening is, in turn, dependent not only on the focal length of the lens, but on 
the size of the lens in relation to the aperture size of the optical system and on the dis¬ 
tance between the lens and the aperture. Thus, if a large lens were placed in front of 
the optical system, only the center (or least deviated part) of the bundle of rays would 
enter. If the large lens were positive and backed off from the optical system, a larger 
portion of the angularly distributed rays would enter the optical system, increasing 
the size of the image in the focal plane. Conversely, if a negative lens were backed 
off from the aperture, a smaller portion of the angularly distributed rays would enter 
the optical system and the size of the focal plane image would decrease and become more 
the size of the undisturbed image. 

The same changes in focal-plane image sizes would result if lenses of shorter and 
longer focal lengths were placed immediately in front of the optical system. Thus, 
there are two mechanisms for producing image-size changes that cannot be differen¬ 
tiated directly. There is a difference between the two mechanisms, however. When 
the focal length of the close-up lens is changed, there is no accompanying change in 
the total radiation in the optical system; when the large lens is backed off, there is a 
change in the total received radiation because a greater portion of the large positive 
lens will be effective in collecting radiated energy to concentrate onto the small receiver 
aperture as the lens is backed off more. Actually, the increase in total received radiated 
energy will continue until one of the following occurs: (1) all of the radiated energy 
falling on the lens is converged onto the receiver aperture, (2) the size of the converged 
beam is limited by the diffraction pattern of the lens, (3) the lens is unable to form an 
image. For the negative lens, more energy will be diverted from the aperture of the 
optical system as the lens is backed off a greater distance. The total energy in this 
case will continue to decrease until the negative lens reaches the midpoint of the 
optical path, or until diffraction around the lens predominates. 

6.6.4. Atmospheric Scintillation Measurements. Data on stellar scintillation and 
scintillation over land and water are given in this section. The data are typical, and 
vast amounts of similar data are given in the references cited. Causes of scintillation 
and its effects on optical systems are discussed in Sec. 6.6. 

Scintillation measurements can be expressed conveniently in terms of percent 
equivalent sine wave modulation per unit bandwidth, which is defined as 


M(%) = lAUEflEac 


(6-51) 


where _Ej = average noise voltage at frequency / 

Edc = average dc level of the signal 

Often, the average percent equivalent sine wave modulation, M(%), is plotted to keep 
the figures for all frequency ranges at about the same value. The average value in a 
given frequency range can be converted to root mean square deviation of the signal in 
that range by multiplying it by the square root of the frequency interval covered. 

The frequency range of scintillation components is approximately 2.5 to 450 cps. 
To simplify scintillation measurements, this frequency range can be broken down 
into subranges, e.g., 2.5 to 105 cps, 10 to 50 cps, and 50 to 450 cps, and expressed as 
M(%) 2 . s-10.5, M(%) 10 -so, and M(%) 5 0 - 4 so. 


ATMOSPHERIC SCINTILLATION 


215 


Scintillation modulus (mod), which is sometimes used to define the shape of the curve, 
is the ratio of the average percent equivalent sine wave modulation in the mid-frequency 
or high-frequency range to that of the low-frequency range. 


modio-5o 


M{%) 10-50 
M(%)2. 5 - 10.5 


mod: 


M(%) 


50-450 


50-450 


M{%) 


2 . 5 - 10.5 


(6-52) 


6.6.4.I. Stellar Scintillation [83, 84]. Figure 6-38 shows a typical plot of percent 
equivalent sine wave modulation, M(%), versus frequency (z = zenith angle). The shape 
of the curve can be characterized by the rollover and crossover frequencies. The 
rollover frequency is the frequency at which an average line drawn through the low- 
frequency components intersects with an average line drawn through the mid- and 
high-frequency components. The crossover frequency is the frequency at which the 
high-frequency average line intersects the zero axis. Determination of the rollover 
and crossover frequencies depends, to a certain extent, on the manner in which the 
scintillation curves are plotted. Figure 6-38, and subsequent stellar scintillation 
curves, are based on semilog plots. 


51 — 


6 s ? 


3 - 


2 - 


— 




— 



Vega 

8/27/53 

12.5" Aperture 00:47 E.S.T. 

Z = 45:4 


Rollover Frequency 
18 cps 

\ l 



Low-Frequency 




Range 

Mid-Frequency 
Range 

High-Frequency 

Range 

Crossover Frequency 140 cps 

i i _ I _1_1_L 

T 1 1 | T i i i | 1 l 1 

I^T 1 I | 1 l 1 | 1 1 1 1 


10 50 100 

FREQUENCY (cps) 


500 1000 


Fig. 6-38. A representative frequency analysis of stellar scintillation [59]. 


Diurnal and Seasonal Variations in Stellar Scintillation. Figure 6-39 is a 
comparison of daytime and nighttime scintillation. The shape of tne daytime and 
nighttime curves are very similar, and the amount of scintillation is only slightly 
greater during the daytime. 













216 


ATMOSPHERIC PHENOMENA 



Fig. 6-39. Comparison of daytime to nighttime stellar scintillation [59]. 


Figure 6-40 shows the typical seasonal variation of stellar scintillation. Total 
scintillation is generally greater in the winter, but the low-frequency components are 
greater during the summer. 

Effects of Upper-Air Winds on Stellar Scintillation. Stellar scintillation 
appears to be virtually independent of surface weather conditions. At the 200-mb 
level (approximately 40,000 ft), however, a correlation exists between the shape of the 
scintillation curve and the wind speed, and also between slit orientation and wind 
direction. 



Fig. 6-40. Comparison of winter to summer stellar scintillation [59]. 
















ATMOSPHERIC SCINTILLATION 


217 


Figure 6-41 shows the relationship between wind speed and the shape of the scintilla¬ 
tion curve at the 200-mb level. Wind speeds are estimated to be accurate within ±10 
knots. The data in Fig. 6-41 were obtained using a 12.5-in. aperture, but the correla¬ 
tion is valid for smaller apertures also. 

Figure 6-42 illustrates the correlation between slit orientation and wind direction. 
The dashed lines represent deviations of ±20°, which are estimates of the probable 
error in reading wind directions from plotted weather maps. 



Fig. 6-41. Relation of scintillation moduli to wind speed at 200-mb level for 
a 12.5-in. aperture [59]. 



Fig. 6-42. Correlation of slit position for minimum 200-cps 
scintillation component with wind direction at 200-mb level [59]. 










218 


ATMOSPHERIC PHENOMENA 


Effects of Zenith Distance on Stellar Scintillation. Stellar scintillation is 
greater near the horizon than it is overhead. It is difficult to precisely describe this 
effect, however, because the manner in which scintillation varies with the secant of the 
zenith angle depends greatly upon aperture size, the frequency range under considera¬ 
tion, and meteorological factors. The latter complicate the determination of zenith 
distance considerably. 

Figures 6-43 and 6-44 show M(%) 2 . 5-10.5 and M(%) total versus log sec Z, respectively, 
for a 12-in. aperture. The same data for a 3-in. aperture are shown in Fig. 6-45 and 
6-46. No meaningful data are available for the mid-frequency scintillation range 
(10 to 50 cps) and the high-frequency range (50 to 450 cps) because of the very strong 
meteorological affects at these frequencies. 

Table 6-5 gives approximate functional relationships between M{%) and sec Z of 
the form a(sec Z) n for various aperture sizes. The values given in the table represent a 
first-order attempt to describe the effect of zenith distance on scintillation in that 
wind-velocity effects are neglected. 

Effects of Aperture Size on Stellar Scintillation. Figure 6-47 shows the 
general manner in which the scintillation curve changes with aperture size. The 
ratio of scintillation at various aperture sizes to scintillation for a 12.5-in. aperture 
is given in Fig. 6-48. In both curves, the relationship between scintillation and aper¬ 
ture is for low wind velocity. The combined effects of both wind velocity and zenith 
distance upon the ratio of the amount of scintillation for a 3-in. aperture and the amount 
for a 12.5-in. aperture is shown in Fig. 6-49 and 6-50. 



Fig. 6-43. Variation of low-frequency stellar scintillation with 
zenith distance for 12.5-in. aperture [59]. 





ATMOSPHERIC SCINTILLATION 


219 



Fig. 6-44. Variation of stellar scintillation over all frequencies 
with zenith distance for 12.5-in. aperture [59]. 



Fig. 6-45. Variation of low-frequency stellar scintillation with 
zenith distance for 3-in. aperture [59]. 









220 


ATMOSPHERIC PHENOMENA 



Fig. 6-46. Variation of stellar scintillation over all frequencies 
with zenith distance for 3-in. aperture [59]. 


Table 6-5. Functional Relationships between Average 
Equivalent Sine Wave Modulation and Zenith Distance [83]. 


12 -in. Aperture 

6-in. Aperture 

2 

, „ . Relationship 

( Degrees) 

% 

, . Relationship 

( Degrees ) 


_ 

65 

1.0 (sec Z) 1 - 8 

60 

1.5 (sec Z) 1 - 6 

M (%) 2.5 — 10.5 

65 

3.6 (sec Z)° 1 

60 

3.6 (sec Z)° 3 

M{%)iotal 

63 

0.3 (sec Z) 15 

58 

0.7 (sec Z) 1 - 2 

1.4 (sec Z)° ° 

63 

1.0 (sec Z)° ° 

58 


3 -in. Aperture 

1-in. 

Aperture 


Z 

( Degrees ) 

Relationship 

Z 

{Degrees) 

Relationship 


49 

2.3 (sec Z) 1 - 5 

48 

2.7 (sec Z) 1 ' 8 

5-10.5 

49 

3.7 (sec Z) 0 - 3 

48 

4.5 (sec Z) 0 - 3 


53 

1.3 (sec Z) 10 

52 

1.7 (sec Z)°- 9 

M(%)ioi a i 

53 

2.4 (sec Z)-°- 2 

52 

3.0 (sec Z)-° 








RATIO 


ATMOSPHERIC SCINTILLATION 


221 



Fig. 6-47. Variation of scintillation curve with aperture [59]. 



Fig. 6-48. Relationship between amount of stellar scintillation for various 
aperture sizes expressed in terms of scintillation from 12.5-in. aperture [59]. 


















222 


ATMOSPHERIC PHENOMENA 



Fig. 6-49. Effect of zenith distance upon ratio of the low-frequency scintilla¬ 
tion components for 3-in. aperture to a 12-in. aperture for different wind 
velocities at 200 mb [59]. 



sec Z 


Fig. 6-50. Effect of zenith distance upon the ratio of the scintillation 
taken over all frequencies for a 3-in. to a 12.5-in. aperture for different 
wind velocities at 200 mb [59], 









ATMOSPHERIC SCINTILLATION 


223 


6.6.4.2. Scintillation Over Water [85, 86]. 

Seasonal Variations. Figure 6-51 shows seasonal variations of atmospheric scintil¬ 
lation over water along a 17,750-yd path. The data were obtained with a 12-in. search¬ 
light mounted 30 ft above the water and a 24-in. optical receiver mounted 109 ft above 
the water. The values of percent equivalent sine wave modulation, M{%), shown in 
Fig. 6-51, are averages for each month. The air temperatures are also average values 
for each month, based on readings obtained at both ends of the transmission path. 

Effects of Receiver Collector Area. Figure 6-52 shows representative frequency 
spectrum curves for four different receiver collector areas. The manner in which 
scintillation varies with receiver collector area is shown in Fig. 6-53. The data are 
based on measurements over water along a 4400-yd path. The transmitting source 
was mounted approximately 18 ft above the water and the receiver approximately 109 
ft above the water. 

Effects of Source Area. Figure 6-54 shows representative frequency-spectrum 
curves for four different source areas. The effective area of each lamp source is approxi¬ 
mately 2.8 in. 2 , and the angle subtended by each lamp is approximately 2.2 seconds of 
arc. Figure 6-55 shows the manner in which total scintillation varies with source 
area. The data in both illustrations were measured over water along a 4400-yd path. 
The transmitting source was mounted approximately 18 ft above the water and the 
receiver approximately 109 ft above the water. 

The amount of scintillation decreases as the number of source lamps (and, consequent¬ 
ly, the area of the source) increase. However, the physical distribution and orientation 
of the source lamps with respect to each other appear to have little effect on the amount 
of scintillation. 

6.6.4.3. Scintillation Over Land [87]. 

Diurnal Variations. Figure 6-56 shows typical diurnal variations in the amount 
of scintillation over a grass surface on a clear day. The data were determined at the 
mid-latitudes near the solar equinox. This pattern of daily variations is subject to 
many influences, but it will retain the general relationship to solar elevation. 

Effects of Temperature and Wind on Scintillation Over Land. In general, the 
amount of scintillation is proportional to the absolute magnitude of temperature 
change with height, although its characteristics depend upon whether temperature 
increases or decreases with height. 

(a) For temperature decreasing markedly with height (an unstable atmosphere), 
the typical condition on a cloudless day (except over snow surfaces), scintillation in¬ 
creases rapidly with increases in temperature gradient but slowly with increase in 
wind speed for winds less than 10 mph. Therefore, under unstable conditions, scintil¬ 
lation is dependent on temperature gradient to a greater degree than on wind speed 
(Fig. 6-57 a). 

( b) For temperature increasing markedly with height (a stable atmosphere), the 
typical condition on a cloudless night, scintillation increases slowly with increases in 
temperature gradient but rapidly with increases in wind speed for winds up to 3 or 
4 mph. Therefore, under stable conditions scintillation is dependent to a greater 
degree on wind speed than on the temperature gradient (Fig. 6-576). 

(c) For adiabatic or near-adiabatic conditions, a decrease in temperature of about 
0.01°C per meter (a neutral atmosphere), scintillation is at a minimum or is absent, 
irrespective of wind. 


224 


ATMOSPHERIC PHENOMENA 


•C 



Fig. 6-51. Seasonal variation of atmo¬ 
spheric scintillation over a water path [62]. 



10 100 1000 1Q000 

FREQUENCY (cps) 


Fig. 6-52. Scintillation vs. frequency 
measured over water [62]. 



Fig. 6-53. Scintillation vs. receiver collector 
area measured over water [62]. 












































































ATMOSPHERIC SCINTILLATION 


225 



FREQUENCY (cps) 

Fig. 6-54. Amount of scintillation over 
water vs. frequency for various source 
areas [62]. 



Fig. 6-55. Scintillation vs. source 
area measured over water [62]. 




























226 


ATMOSPHERIC PHENOMENA 


>* 

c o 
Z 

w 

H 


u 

E 

H 

< 

►J 

W 

a 


16u 


12 - 


4- 


00 


<L> 

CO 

•H 

Sh 

a 

o 

co 


/ 


CD 

C/3 

e 

d 

co 


\ / 
» / 
V/ 


A 

1 ' A 

/ v '_••• 


04 


08 


12 


16 


20 


24 


TIME RELATIVE TO SUN (hr) 

Fig. 6-56. Diurnal cycle of scintil¬ 
lation over a grass surface [83]. 


Vertical 

Temperature 

Difference 



Fig. 6-57. Percent modulation as a 
function of temperature difference 
and wind speed (temperature dif¬ 
ference between 0.5-m height; wind 
speed measured at 2-m height) [83]. 











SOLAR SPECTRUM MEASUREMENTS 


227 


Figure 6-58 shows the effects of an unstable, a stable, and a neutral atmosphere on 
the frequency of atmospheric scintillation. Three distinct conditions are illustrated: 
(1) midafternoon, with strong lapse conditions following the time of maximum heating 
of the ground; (2) the sunset period, during which the air is close to neutral equilibrium; 
(3) at night, after an inversion has formed because of surface cooling. 



a> 

T3 

3 


a 

S 

< 

a> 

> 


.2 

”3 

os 


Fig. 6-58. Effects of a stable, unstable, and neutral 
atmosphere on atmospheric scintillation over a grass 
surface [83]. 


6.7. Solar Spectrum Measurements 

A solar spectrum is a transmission spectrum of the earth’s atmosphere in which 
the sun is used as the source of radiation. Many Fraunhofer lines are observed in the 
visible and ultraviolet portions of the solar spectrum. In the infrared, however, very 
few Fraunhofer lines are present and the sun emits approximately as a uniform 6000°K 
blackbody. Spectral details observed in the infrared solar spectrum are almost entirely 
due to the absorption of solar radiation by the molecules present in the earth’s atmos¬ 
phere. 

The length of an absorption path through the atmosphere is dependent upon the 
elevation of the sun in the sky. Thus, when the sun is at the zenith, the solar radiation 
traverses one air mass of atmosphere. At any other angle from the zenith, called the 
solar altitude or zenith distance, the radiation traverses longer paths through the 
atmosphere. 

Table 6-6 gives the equivalent air mass for a sea-level observer as the solar altitude 
varies from 0° to 90° [89]. To obtain the same type of table for an observer at a dif¬ 
ferent reference altitude, multiply the values in Table 6-6 by the ratio of the pressure 
at the new reference level to that at sea level. This approximation holds well up to 
85° angles and 100-km altitude. 

Figure 6-59 shows a low-resolution solar spectrum for the region from 1 to approxi¬ 
mately 15 /x. The other curves show the position and approximate relative intensities 
of the infrared absorption bands for various molecules in the atmosphere. 

A number of high-resolution measurements of the solar spectrum have been made 
[90-103]. Figures 6-60 through 6-74 show the spectrum from approximately 2.80 to 
14.2 /x [90-93]. The measurements were made from the Jungfraujoch, Switzerland, 
at an altitude of approximately 12,000 ft. Figures in the references contain more detail. 

In the region from 2.80 to 3.15 /x (Fig. 6-60) the absorption is due mainly to H 2 0, 
although lines of a band 13co 2 , centered at 2.834 /x, v\ -I- v 3 of N 2 0, centered at 2.87 /x, 
and 2v 2 + v 3 of N 2 0, centered at 2.97 /x, are present. In the region of 3.15 to 3.50 /x 
(Fig. 6-61), the strong fundamental band v 3 of CH 4 , centered at 3.31 /x, is present. The 
remaining absorption is due to the weaker 2v 2 band of H 2 0. 




228 


ATMOSPHERIC PHENOMENA 


Table 6-6. Equivalent Air Masses for Solar Altitudes 0° to 90° a 



0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

0° 

— 

26.96 

19.79 

15.36 

12.44 

10.40 

8.90 

7.77 

6.88 

6.18 

10° 

5.60 

5.12 

4.72 

4.37 

4.08 

3.82 

3.59 

3.39 

3.21 

3.05 

to 

o 

o 

2.90 

2.77 

2.65 

2.55 

2.45 

2.36 

2.27 

2.20 

2.12 

2.06 

o 

O 

CO 

2.00 

1.94 

1.88 

1.83 

1.78 

1.74 

1.70 

1.66 

1.62 

1.59 

O 

o 

1.55 

1.52 

1.49 

1.46 

1.44 

1.41 

1.39 

1.37 

1.34 

1.32 

cn 

O 

0 

1.30 

1.28 

1.27 

1.25 

1.24 

1.22 

1.20 

1.19 

1.18 

1.17 

60° 

1.15 

1.14 

1.13 

1.12 

1.11 

1.10 

1.09 

1.09 

1.08 

1.07 

o 

o 

1.06 

1.06 

1.05 

1.05 

1.04 

1.04 

1.03 

1.03 

1.02 

1.02 

80° 

1.02 

1.01 

1.01 

1.01 

1.01 

1.00 

1.00 

1.00 

1.00 

1.00 

90° 

1.00 

_ 

_ 

_ 

_ 

_ 

_ 

_ 

_ 

— 


“Entries in the table are the air masses for the angles indicated down the left and across the top 
For example the air mass for 22° elevation is 2.65. 


0 

100 

. co 

0 

100 


CH. 

4 

_i_i_i-1 ‘ — 1 - 

T T 

1_ 

O O 

O 

v—* 

e 

..i 

. .11 

N 2° 

TION (% 

>—* 

O 

O O 

i_i_i_i_i_i_i 

7 V 

°3 

_1_1_1-1 i- 

4BSORF 

o 

O Ol 

~T\ 

a 

.V. C °2 

0 

100 

Y 

HDO 

-1_* 1 1 _1_1_1_1_1_1_1_» * 

0 

100 


r 


H 2° 

0 

100 

to 

jA A .JA 

n 


£ 

000 ,3000 2000.160014001200 1000 900 800 700 cm- 1 


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 u 


Fig. 6-59. Low-resolution solar spectrum from 1 to 24 g. 














































SOLAR SPECTRUM MEASUREMENTS 


229 


From 3.50 to 3.85 fx (Fig. 6-62), many strong lines of the v x fundamental band of 
HDO are present, in additio? to some CH 4 absorption (u 2 + v 4 at 3.55 /x) and a weak 
N 2 0 combination band (v 2 + v 3 at 3.57 /x). The Q-branch of the HDO band appears 
as a weak cluster of lines near 3.67 fx. All lines marked X in Fig. 6-62 are due to HDO. 

The region from 3.85 to 4.20 \x (Fig. 6-63) contain 2 v x of N 2 0 at 3.91 /x and v x + 2v 2 
of N 2 O at 4.06 fx . It also contains the v 3 fundamental of C0 2 , beginning at 4.18 /z. 

Virtually complete absorption by C0 2 occurs in the region from 4.19 /x (Fig. 6-63) 
to 4.45 fx (Fig. 6-64). From 4.43 to 4.48 fx are some high-J lines in the P-branch of 
V 3 of 13 C0 2 , centered at 4.38 fx. The intense u 3 fundamental band of N 2 0 is at 4.49 /z, 
with a weaker band {v 3 -f v 2 ) — v 2 of N 2 0 centered near 4.52 /z. 

Near 4.66 /z (Fig. 6-64) is the fundamental band of CO. Circles above the lines in 
(Fig. 6-64) indicate CO transitions of solar origin. The absorption at approximately 
4.7 /z is due to ozone. The ozone absorption lines are indicated by circles below the 
spectrum. H 2 0 absorption lines between about 4.64 and 4.68 ix are indicated by H. 

A weak C0 2 band is present near 4.8 /x (Fig. 6-65) and a very weak C0 2 band near 
5.2 fx (Fig. 6-66). Beyond about 5.2 /z, there is strong absorption from the edge of the 

fundamental band of H 2 0 which is centered at 6.2 fx. Absorption by water vapor is 
complete between 5.5 and 6.9 /x. From 6.9 to 7.65 ix (Fig. 6-67), the primary absorption 
is the edge of the 6.2 -/x H 2 0 band, although absorption due to the v 4 vibrational-rotation 
band of CH 4 , centered at 7.65 /x, is present. The region between 7.65 /z (Fig. 6-68) and 
9.0 ix (Fig. 6-69) possesses relatively high transmission, except for the overtone band 
2 v 3 of N 2 0 centered at 8.56 /z. 

Figure 6-70 shows very intense absorption due to the v 3 band of 0 3 centered at 9.60 /z. 
The very faint structure observed between 8.90 fx (Fig. 6-69) and 9.15 fx (Fig. 6-70) is 
due to the very weak v x band of 0 3 . 

The v 3 — 2v 2 difference band of C0 2 , centered at 9.4 /x, is also shown in Fig. 6-70. In 
a difference band, the individual absorption line is not caused by a quantum transition 
from the ground state of the molecule to an excited level but rather to a transition from 
an excited level to a higher excited level. Because the intensity of an absorption line 
depends very strongly on the population of the energy level from which the transition 
originates, and since this population decreases as temperature decreases, difference 
bands fade out very rapidly as temperature decreases. Thus, a band such as the 9.40-/Z 
C0 2 band, although it may cause significant absorption near ground level where the 
ambient temperature may be 300°K, may have very little intensity near 100,000 ft 
where the temperature is about 200°K. 

The region from 9.75 to 10.6 /x (Fig. 6-71) shows more of the structure of the 9.6-/x 
ozone band. It also shows another difference band of C0 2 , v 3 — v 1 , centered at 10.4 /z. 

Relatively high transmission is present in the region from 12.2 to 13.25 /x (Fig. 6-72 
and 6-73). Figure 6-73 shows a C0 2 difference band, v x — v 2y centered near 13.2 /z and a 
much weaker C0 2 difference band, (v x + v 2 ) — 2 v 2 , centered near 12.6 /x. 

Figure 6-74 shows more of the structure of the v x — v 2 difference band of C0 2 and, 
starting near 13.7 ix, the absorption by the very intense v 2 fundamental band of C0 2 . 
Absorption by this band is complete to about 17 /z. 




230 


ATMOSPHERIC PHENOMENA 



Fig. 6-60. Solar spectrum from 2.80 to 3.15 pi (lowest curve); 
laboratory spectrum of H 2 0 (top curve); laboratory spectrum of 
N 2 0 (middle curve). 





Fig. 6-61. Solar spectrum from 3.15 to 3.50 pi (lower curve); laboratory spectrum of CH 4 
(upper curve). 























































































SOLAR SPECTRUM MEASUREMENTS 


231 






Wavelength 


Fig. 6-62. Solar spectrum 3.50 to 3.85 /u, (lower curve); laboratory spectrum of CfL (lower 
curve). 


2 v. N„0 



^"^^^vwv^/VvvvwvvVWVWVVV^ —"^/'A/yvvvVWVYVVVVYVVVVwvv^w^ —■— 


3.85 3.90 3.95 

I_I_I_I_I_I-1-1-1-1_I_L 

N 2° 



4.00 

j|I_I 


4.05 

I_I_I_L 


MAAAAAAAAAAAAAAAaaaa^^^^^^ 



Fig. 6-63. Solar spectrum from 3.85 to 4.19 /u. (lower curve); laboratory spectrum of N 2 0 
(upper curve). 






















232 


ATMOSPHERIC PHENOMENA 



Wavelength 


Fig. 6-64. Solar spectrum from 4.43 to 4.73 /x. Absorption structure by C0 2 , N 2 0, and C0 2 
are shown schematically. 





Fig. 6-65. The solar spectrum from 4.70 to 5.11 /a. 























































































































































SOLAR SPECTRUM MEASUREMENTS 


233 




Wavelength 


Fig. 6-66. The solar spectrum from 5.11 to 5.55 /x. 



6.90 6.95 7.00 7.05 7.10 7.15 

I i i i i I i i i i I _I_I_I_I-L—1-1—I-1-1-1-1-1-1-1-1-1-L 



Wavelength 


Fig. 6-67. Solar spectrum from 6.90 to 7.65 /x (lowest curve); laboratory spectrum of H 2 0 
(top curve); laboratory spectra of CH4 (middle curve). 





























234 


ATMOSPHERIC PHENOMENA 




Wavelength 

Fig. 6 -68. Solar spectrum from 7.65 to 8.35 /a (lower curve); laboratory spectrum of CH 4 
(upper curve); absorption structure of N 2 0 indicated schematically. 


Band 


R30 


R 20 


RIO 


Center 



P 40 


P 49 



Fig. 6-69. Solar spectrum from 8.35 to 9.03 /a. Absorption structure of N 2 0 indicated 
schematically. 




































































































SOLAR SPECTRUM MEASUREMENTS 


235 


R40 R30 

iiiii i : 


R 20 

I 



R 20 


RIO 


R 2 


Band 

Center p 2 

' f I 

"3 - ^2 
CO„ 


P10 



9 30 

1 1 1 I I ■ 1 1 9 '. 35 .... 9 '. 40 . ■ ■ ■ 9 '. 45 ■ ■ , 


9.50 


P 20 P 30 

I I I I I 1 


I I I 


P 40 


0„ 


■ ■ ■ ■ »,»» ■ ■ ■ ■ »•■«> ■ , ■ ■ -,«» , ■ ■ , »-.™ , , , 


Wavelength 


Fig. 6-70. Solar spectrum from 9.03 to 9.73 /a. Absorption structure of C0 2 indicated 
schematically. 


VvAMMM/WAvAaaaAa/\w^^ 

■ ■ . 9 -? 5 .... 9 -. 80 .... 9 -. 65 , , , , 9 -?° I ,,, 9 - 95 


R 40 R 30 R 20 

1 1 < M I | | | | | j 




Wavelength 


Fig. 6-71. Solar spectrum from 9.72 to 10.7 /x. Absorption spectra of C0 2 indicated 
schematically. 


















236 


ATMOSPHERIC PHENOMENA 


P 30 

I l 


P 40 

i l l 



11.1 

-L_ 


- ^- 

11.2 11.3 

J_I-1-L 


"V- 

11.4 

L 


V 



R40 R32 

I I I I I 



Wavelength 

Fig. 6-72. Solar spectrum from 10.7 to 12.2 /*. Absorption structure of C0 2 indicated 
schematically. 



Wavelength 


Fig. 6-73. Solar spectrum from 12.2 to 13.3 /x. Absorption structure for C0 2 indicated 
schematically. 

































TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 237 



Fig. 6-74. Solar spectrum from 13.25 to 14.2 /x. Absorption structure of C0 2 indicated 
schematically. 


6.8. Total Absorption (Laboratory Measurements) [104-107] 


Total absorption data provide a means to predict absorption for known paths through 
known absorbing gases and to test the validity of theories describing absorption phe¬ 
nomena. The total absorption of an absorption band is the area under the curve 
obtained when the fractional absorption at a given frequency is plotted against fre¬ 
quency. Usually the integral 



A(v ) dv 


is called the total absorption of an absorption band, defined by the limits v\ and v 2 , and 
is expressed in frequency units. Sometimes, however, the integral is referred to as the 
equivalent bandwidth of the absorption because the same integral can be considered as 
applying to an equivalent band having complete absorption over a frequency interval 


r v t 

Av = I A(v) dv 

i 


The former definition is used in this section. 

Data presented in this section show the functional relationship between total absorp¬ 
tion fA(v)dv and absorber concentration w, partial pressure of the absorbing gas p, 
total pressure P, which includes the partial pressure of absorbing and nonabsorbing 
gases, and the absolute temperature of the gas T. This relationship is expressed as 



A(v) dv = <j>(w,p,P, T) 


and is described for various wavelength regions of high characteristic absorption. 

The data were measured using multiple-traversal cells containing absorbing gas the 
partial pressure of which could be varied. High-altitude conditions were simulated by 
























































238 


ATMOSPHERIC PHENOMENA 


proper vacua by adding broadening gases such as nitrogen and oxygen to the cell. 
Path lengths from 1.5 to 48 m were achieved by successive reflections of radiation back 
and forth through the cell. 

6.8.1. Total Absorption by C0 2 . Strong absorption by C0 2 exists in the 2.7-/n 
(3660-cm -1 ) region, the 4.3-/z (2350-cm -1 ) region, and the region between 11.4 fx (875 
cm -1 ) and 20 /x (495 cm -1 ). Weaker absorption bands are present at 1.4 /x (6975 cm -1 ), 
1.6 [x (6230 cm -1 ), 2.0 /z (4983 cm -1 ), 4.8 (x (2075 cm -1 ), 5.2 /z (1930 cm -1 ), 9.4 /z (1064 
cm -1 ), and 10.4 fx (961 cm -1 ). In the figures that follow, P e = equivalent pressure as 
used by Burch and coworkers, and P is total pressure. 

6.8.1.1. The 2.7-/z (3660-cm -1 ) Region. The C0 2 absorption in the 2.7 -/x region is 
caused primarily by two strong absorption bands, the 2v 2 + v 3 band centered at 2.77 /z 
(3609 cm -1 ) and the v\ + v 3 band centered at 2.69 /z (3716 cm -1 ). Total absorption for 
the 2.77-/Z (3609-cm -1 ) band is shown in Fig. 6-75, and total absorption for the 2.69-fx 
band (3716 cm -1 ) band in Fig. 6-76. Total absorption for the entire 2.7-/z (3660 cm -1 ) 



Fig. 6-75. Total absorption vs. absorber concentration 
of the 2.77-/X (3609-cm -1 ) C0 2 band. 



Fig. 6-76. Total absorption vs. absorber concentration for 
the 2.69 -fj. (3716-cm -1 ) C0 2 band. 






TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


239 



Fig. 6-77. Total absorption vs. absorber concentration for the 2.7-p. (3660-cm -1 ) C0 2 
region (2.77-/X band + 2.69-/X band). 


region, i.e., the 2.77-/X (3609-cm -1 ) band plus the 2.69-/X (3716-cm -1 ) band, is shown in 
Fig. 6-77. For a total absorption of more than 3 cm' 1 , the curves in all of the illustra¬ 
tions are estimated to be accurate within ±5%; between 3 and 10 cm -1 , the estimated 
accuracy is ±10%. For total absorption values less than 3 cm -1 , the estimated accuracy 
is ±20%. 

6.8.I.2. The 4.3-fx (2350-cm -1 ), 4.8-/x (2075-cm -1 ), and 5.2-fx (1930-cm -1 ) Bands. 
The 4.3 -fx (2350-cm -1 ) C0 2 band causes almost complete absorption between about 
4.19 fx (2386 cm -1 ) and 4.45 fx (2250 cm -1 ). Total absorption curves, which are esti¬ 
mated to be accurate within ±5% above 10 cm -1 total absorption, ±10% between 3 and 
10 cm -1 total absorption, and 20% below 3 cm -1 total absorption, are shown in Fig. 6-78. 

Total absorption curves for the 4.8-fx (2075-cm -1 ) and the 5.2 -/x (1930-cm -1 ) band are 
shown in Fig. 6-79 and 6-80, respectively. These bands are very weak and are sig¬ 
nificant only for high values of absorber concentration. 



Fig. 6-78. Total absorption vs. absorber concentration for the 4.3-p. (2350-cm ') C0 2 band. 






240 ATMOSPHERIC 



w (atm cm) 

Fig. 6-79. Total absorption vs. absorber 
concentration for the 4.8 -/a (2075-cm 1 ) C0 2 
band. 


PHENOMENA 



w (atm cm) 


Fig. 6-80. Total absorption vs. absorber 
concentration for the 5.2-/x (1930-cm -1 ) C0 2 
band. 



Fig. 6-81. Total absorption vs. absorber 
concentration for the 9.4-/a (1064-cm 1 ) 
C0 2 band. 



Fig. 6-82. Total absorption vs. absorber 
concentration for the 10.4-/X (961-cm -1 ) 
C0 2 band. 


6.8.1.3. The 9.4 -fx (1064-cm -1 ) and 10.4-/x (961-cm -1 ) Bands. Total absorption of 
the weak C0 2 bands at 9.4 /x (1064 cm -1 ) and 10.4 /x (961 cm -1 ) are shown in Fig. 6-81 
and 6-82, respectively. Total absorption of these bands is strongly dependent upon 
temperature; thus, values of total absorption are temperature corrected. The curves 
shown in Fig. 6-81 and 6-82 are for a temperature of 26°C, and are estimated to be accu¬ 
rate within ±5% for more than 10-cm -1 total absorption greater than 10 cm -1 , ±10% 
between 3 and 10-cm -1 total absorption, and ±20% below 3 cm -1 total absorption. 

The temperature dependence of both bands is illustrated in Fig. 6-83. From the 
curves, it can be seen that total absorption increases with temperature. 

6.8.1.4. The 1.4-p, (7150-cm -1 ), 1.6-/x (6250-cm -1 ), and 2.0-/x (5000-cm -1 ) Bands. 
Total absorption of the very weak C0 2 bands at 1.4 /x (7150 cm -1 ), 1.6 /x (6250 cm -1 ) 
and 2.0 /x (5000 cm -1 ) are shown in Fig. 6-84 through 6-86. The band at 1.6 fx (6250 
cm -1 ) is actually a group of very weak bands centered at 1.645 /x (6077 cm -1 ), 1.604 /x 
(6231 cm -1 ), 1.574 /x (6351 cm -1 ), and 1.536 /x (6510 cm -1 ). The C0 2 band near 2 /x 
(5000 cm -1 ) consists primarily of three weak absorption bands centered at 2.057 /x 
(4861 cm -1 ), 2.006 /x (4983.5 cm -1 ), and 1.957 /x (5109 cm -1 ). 











TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


241 



20 40 60 80 

TEMPERATURE (°C) 

Fig. 6-83. Effects of temperature on 
total absorption for the 9.4-p, (1064- 
cur 1 ) and 10.4-p. (961-cur 1 ) C0 2 bands. 



w (atm cm) 

Fig. 6-84. Total absorption us. absorber 
concentration for the 1.4-/x (7150-cur 1 ) 
C0 2 band. 



Fig. 6-85. Total absorption us. absorber 
concentration for the 1.6-/x (6250-cnr 1 ) 
C0 2 band. 



Fig. 6-86. Total absorption us. 
absorber concentration for the 
2-/u. (5000-cur 1 ) C0 2 band. 








242 


ATMOSPHERIC PHENOMENA 


6.8.I.5. The 11.4-g (875-cm -1 ) to 20-g (495-cm -1 ) Region. There are several strong 
and medium C0 2 absorption bands in the spectral region between 11.4 g (875 cm -1 ) 
and 20 g (495 cm -1 ), the strongest of which is centered at 14.9 g (667 cm -1 ). Because 
of the wide spectral region covered in this section, the spectral region is divided into 
smaller subregions, and absorber concentration is plotted against mean fraction ab¬ 
sorption rather than against total absorption, which usually refers to an entire band. 
Total absorption / A(v)dv of an entire band is independent of spectral slit width, pro¬ 
vided that there is no absorption beyond the limits of integration. It follows that the 
total absorption of a subregion will be independent of spectral slit width if the sub- 
regions are divided at frequencies where the absorption is zero. Practically, however, 
no frequencies exist where the absorption is zero, and optimum frequencies, i.e., fre¬ 
quencies at which absorption is very slight, are chosen to divide the subregions. 

Mean fractional absorption is related to total absorption by the following equation: 


A{v i — v-i) 


1 

(v 2 — Vl) 



dv 


(6-53) 


which represents the mean fractional absorption in the spectral region v x — v 2 . 

Figures 6-87 through 6-91 show mean fractional absorption for each of the five 
subregions of the 11.4-g (875-cm 1 ) to 20.2 g (495-cm -1 ) C0 2 region. The curves are 
based upon a temperature of 26°C, and are estimated to be accurate within ±5% for a 
mean fractional absorption greater than 0.10 cm -1 , and increase to approximately ±20% 
for smaller values of total absorption. 

The effect of temperature on mean fractional absorption is shown in Fig. 6-92 for 
four of the five subregions. No data are available for the fifth subregion, which ex¬ 
tends from 18.3 g (545 cm -1 ) to 20 g (495 cm -1 ). From the curves in Fig. 6-92, it can 
be seen that mean fractional absorption increases with temperature. 



Fig. 6-87. Mean fractional absorption vs. absorber concentration for 11 A-g (875-cm -1 ) 
to 13.9-/i (720-cm -1 ) C0 2 band. 





TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


243 


's 

o 


in 

t> 

00 

I 

o 

CSJ 


k 



Fig. 6-88. Mean fractional absorption vs. absorber concentration for the 13.9-pi (720-cnrr 1 ) 
to 14.9-/x (667-cm -1 ) C0 2 band. 



Fig. 6-89. Mean fractional absorption vs. absorber concentration for the 14.9 -/t (667-cm *) 
to 16.2-p. (617-cm -1 ) C0 2 band. 



Fig. 6-90. Mean fractional absorption vs. absorber concentration for the 16.2-p. 
(617-cm- 1 ) to 18.3-/x (545-cm- 1 ) C0 2 band. 













244 


ATMOSPHERIC PHENOMENA 



Fig. 6-91. Mean fractional absorption vs. absorber 
concentration for the 18.3-p (545-cm -1 ) to 20-p (495- 
cm -1 ) C0 2 band. 


o 

c- 

i 

c~ 

CD 

CD 




I 



TEMPERATURE (°C) 


Fig. 6.92. Effects of temperature on mean fractional ab¬ 
sorption of C0 2 from 11.4 p (875 cm' 1 ) to 18.3 p (545 cm -1 ). 


6.8.2. Total Absorption by H 2 0 [106, 107]. Total absorption of the 6.27-p (1595- 
cm -1 ), 2.70-p (3700-cm -1 ), and 1.87-p (5332-cm -1 ) H 2 0 bands are shown in Fig. 6-93 
through 6-95. The absorption in the 2.70-p region is caused primarily by two absorp¬ 
tion bands with centers at 2.73 p (3657 cm -1 ) and 2.66 p (3756 cm -1 ). The curves in 
Fig. 6-93 through 6-95 are estimated to be accurate within ±6%. 

Minor absorption bands at 3.2 p (3150 cm -1 ), 1.38 p (7250 cm -1 ), 1.1 p (8807 
cm -1 ), and 0.94 p (10,613 cm -1 ) are shown in Fig. 6-96 through 6-99. These curves 
are estimated to be accurate within about ±10%. 
















TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 



Fig. 6-93. Total absorption vs. absorber concentration for the 6.27-fj. 
(1595-cm _1 ) H 2 0 band. 



Fig. 6-94. Total absorption vs. absorber concentration for the 2.70-/x (3700-cm -1 ) 
H 2 0 band. 



245 


Fig. 6-95. Total absorption vs. absorber concentration for the 1.87-/i (5332-cm- 1 ) 
H 2 0 band. 








246 


ATMOSPHERIC PHENOMENA 



Fig. 6-96. Total absorption vs. absorber con¬ 
centration for the 3.2-// (3150-cmr 1 ) H 2 0 band. 




Fig. 6-97. 


Total absorption vs. absorber concentration for the 1.38-//. (7250-cm *) H 2 0 band. 



Total 



Fig. 6-98. Total absorption vs. 
absorber concentration for the 
1.1-// (8807-cm -1 ) H 2 0 band. 


Fig. 6-99. Total absorption vs. 
absorber concentration for the 
0.94-// (10,163-cm- 1 ) H 2 0 band. 


6.8.3. Total Absorption by N 2 0 [106, 107], Figure 6-100 shows total absorption 
as a function of absorber concentration and equivalent pressure for the v 3 fundamental 
band of N 2 0 centered at 4.5 // (2224 cm -1 ). This band occurs in the atmospheric 
window between the strong 4.3-// (2350-cm -1 ) C0 2 band and the 6.2-// (1595-cm -1 ) 
H 2 0 band. Thus, it gives rise to the major portion of the atmospheric absorption in 
the 4.3-6.2-// window. It is estimated that the accuracy of the total absorption values 
given in Fig. 6-100 is within ±5% below 10 cm -1 total absorption and within ±5% above 
10 cm -1 total absorption. 

Figures 6-101 through 6-105 show total absorption of N 2 0 in the 3.9-// (2563-cm -1 ), 
4.05-// (2461-cm -1 ), 7.7-// (1285-cm -1 ), 8.6-// (1167 cm -1 ), and 17.1-// (589 cm -1 ) bands. 
These curves show total absorption within an estimated accuracy of ±5% above a total 
absorption of 30 cm; below 30 cm -1 , the accuracy is somewhat less. 












TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


247 



Fig. 6-100. Total absorption vs. absorber concentration for the 4.5- yu. (2224-cm 1 ) N 2 0 band. 



Fig. 6-101. Total absorption vs. absorber concentration 
for the 3.9-/x (2563-cm 1 ) N 2 0 band. 



Fig. 6-102. Total absorption vs. absorber concentration 
for the 4.05-^t (2461-cm -1 ) N 2 0 band. 









248 


ATMOSPHERIC PHENOMENA 



Fig. 6-103. Total absorption vs. absorber concentration for the 
7.7-/x (1285-cm -1 ) N 2 0 band. 



Fig. 6-104. Total absorption vs. absorber concentration for 8.6-/n (1167-cm -1 ) 
N 2 0 band. 










TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


249 



Fig. 6-105. Total absorption vs. absorber concentration for the 17.1-/X (589-cm 1 ) 
N 2 0 band. 


6.8.4. Total Absorption by CO. Total absorption of the 4.6-/x (2143-cm -1 ) CO band 
is shown in Fig. 6-106. The data are estimated to be accurate within ±5% above 
10-cm -1 total absorption and within ±10% below 10-cnr 1 total absorption. 

Figure 6-107 shows total absorption for the 2.3 -/x (4260-cm -1 ) CO band. The curves 
in Fig. 6-107 do not represent nearly as wide a range of absorber concentration and 
equivalent pressure as those of Fig. 6-106 because the largest values of equivalent 
pressure were not sufficiently large, nor the smallest values of absorber concentration 
sufficiently small, to produce complete absorption over the bands. By comparing Fig. 
6-106 and 6-107, however, it can be seen that, at any given equivalent pressure, the total 



Fig. 6-106. Total absorption vs. absorption concentration for the 4.6-/x (2143-cm *) CO band. 






250 


ATMOSPHERIC PHENOMENA 



10 40 100 400 1000 

w (atm cm) 

Fig. 6-107. Total absorption vs. absorber concentration 
for the 2.3-/x (4260-cm -1 ) CO band. 


absorption of the 4.6-p, (2143-cm -1 ) band is very nearly the same as that of the 2.3-/U, 
(4260-cm -1 ) band at the same equivalent pressure but with the value of the absorber 
concentration 150 times as great. Thus, since the line shape, line spacing, and relative 
line strength within the bands are similar, the total absorption of the weaker over¬ 
tone band (Fig. 6-107) can probably be determined from the curves of the fundamental 
band (Fig. 6-106) by use of the same value of equivalent pressure and a value of ab¬ 
sorber concentration that is 1/150 that of the absorber whose total absorption is being 
measured. 

6.8.5. Total Absorption by CH 4 . Figure 6-108 shows the total absorption of the 
3.31 -ix (3020-cm -1 ) CH 4 band. The values of total absorption are estimated to be ac¬ 
curate within ±5% above 10-cnrr 1 total absorption and within ±10% below 10-cm 1 total 
absorption. 



Fig. 6-108. Total absorption vs. absorber concentration for the 3.3-/x (3020-cm 1 ) CH 4 band. 






TOTAL ABSORPTION (LABORATORY MEASUREMENTS) 


251 


Figures 6-109 and 6-110 show the total absorption of the 7 .6-fi (1306-cm -1 ) CH 4 band 
and the 6.5-/U, (1550-cm -1 ) CH 4 band, respectively. Actually, absorption by the 7.6-/x 
(1306-cm -1 ) band overlaps the absorption by the 6.5-/x (1550-cm -1 ) band for high ab¬ 
sorber concentrations. The estimated accuracy of the total absorption given for the 
7.6-fx (1306-cm -1 ) band (Fig. 6-109) is the same as that for the 3.31-/A (3020-cm -1 ) 
band (Fig. 6-108). Because of overlapping by water vapor, however, the curves for the 
6.5-/x (1550-cm -1 ) band (Fig. 6-110) are estimated to be accurate to no better than 
±10% for total absorption values greater than about 30 cm -1 and ±20% for total absorp¬ 
tion values less than about 30 cm -1 . 



Fig. 6-109. Total absorption vs. absorber concentration for the 7.6-/a (1306-cm *) CH 4 band. 



Fig. 6-110. Total absorption vs. absorber con¬ 
centration for the 6.5-fi (1550-cm -1 ) CH 4 band. 






252 ATMOSPHERIC PHENOMENA 

6.9. Infrared Transmission Through the Atmosphere 

6.9.1. Horizontal Paths [108]. Sea-level transmission measurements over paths 
of 0.3, 5.5, and 16.25 km are shown in Fig. 6-111 through 6-119. The 0.3-km path 
measurements cover the spectral range from approximately 0.5 to 26 /x with a resolution 
ranging from 1 to 2 wave numbers in the regions beyond 2 /x. The 5.5- and 16.5-km 
paths cover the spectral range from approximately 0.5 to 15 /lx with an average re¬ 
solving power, X/AX, of about 300. 

Transmission measurements at 10,000 ft over a path of 27.25 km are shown in 
Fig. 6-120 and 6-121. The spectral range from approximately 0.5 to 15 /x is covered 
with an average resolving power, X/AX, of about 300. 

"Selective-window transmission” from 0.94 to 15 /lx is shown in Fig. 6-122. A window 
is a region of relatively high transmission between absorption bands (Table 6-7). 
For a given window, transmission is the ratio of the energy that actually penetrates 
the atmosphere between the limits of the window to the energy that would be received in 
the absence of any selective absorbers. Thus, it accounts for only selective absorption, 
and does not take scattering losses into consideration. 

The dashed lines in Fig. 6-122 represent the slope of the approximate transmission 
curves described in [109]. 

Windows I and IX (Table 6-7) are not plotted because window I is only slightly de¬ 
pendent on water-vapor concentration and window IX becomes essentially opaque 
through only 5 or 6 mm of water vapor. 




Fig. 6-111. Atmospheric transmission at sea level over a 0.3-km path, 
0.5 to 2.8 /x [108], 





























INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


253 




4.3 4.5 5.0 5.5 

WAVELENGTH (p) 

Fig. 6-112. Atmospheric transmission at sea level over a 0.3-km path, 
3.0 to 518.5 m [108]. 


100 

90 


0 8 ° 

Z 70 
O 

£ 60 


Z 40 
t 30 


20 

10 

0 



5.5 


5.7 mm Precipitable Water 
79°F 


» 2 ° 


1■ ■■ ■ I ■i i i I 

6.0 6.5 7.0 

WAVELENGTH (p) 



Fig. 6-113. Atmospheric transmission at sea level over a 0.3-km path, 
5.5 to 8.5 m [108], 


































































































254 


ATMOSPHERIC PHENOMENA 




Fig. 6-114. Atmospheric transmission at sea level over a 0.3-km path, 
8.5 to 14.0 fji [108]. 




Fig. 6-115. Atmospheric transmission at sea level over a 0.3-km path, 
15 to 25 /x [108]. 


















































TRANSMISSION (%) 


INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


255 



Fig. 6-116. Atmospheric transmission at sea level over 5.5- and 16.25-km paths, 
0.5 to 4.0 [108]. 



Fig. 6-117. Atmospheric transmission at sea level 
over 5.5- and 16.25-km paths, 4.5 to 5.5 [108]. 




5.5 km 

16.25 km 

R.H. (%) 

51 

53 

Temp. (°F) 

64 

68 

H 2 O in path (cm) 

4.18 

15.1 

Transmission 
at 0.55 p (%) 

70 

43 


Fig. 6-118. Atmospheric transmission at 
sea level over 5.5- and 16.25-km paths, 8.0 
to 14 pi [108]. 


















TRANSMISSION (%) 


256 


ATMOSPHERIC PHENOMENA 




WAVELENGTH (/j) 

Fig. 6-119. Atmospheric transmission at sea level over a 16.25-km path 
[108], 



g 100 
80 
60 
40 
20 
0 


Z 

o 

►—. 

co 

co 


co 

Z 

< 

OS 

H 



WAVELENGTH ( 4 ) 


Fig. 6-120. Atmospheric transmission at 10,000 ft over a 27.7-km path 
when 0.55-/U. transmission is 26.5% [108]. 


100 

80 

60 

40 

20 

0 

0, 



WAVELENGTH ( 4 ) 

Fig. 6-121. Atmospheric transmission at 10,000 ft over a 27.7-km path 
when 0.55-/x transmission is 41% [108]. 
























Selective Window Selective Window Selective Window 

Transmission (%) Transmission (%) Transmission (%) 


INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


257 



Precipitable Water Vapor in Path 
II (0.94-1.13 p) 



Precipitable Water Vapor in Path 
IV (1.38-1.90 p) 



Precipitable Water Vapor in Path 
III (1.13-1.38pt-) 



Precipitable Water Vapor in Path 
V (1.90-2.70 p) 



Precipitable Water Vapor in Path 
VI (2.70-4.3p) 



Precipitable Water Vapor in Path 
VII (4.3-6.0 p) 



Precipitable Water Vapor in Path 
VIII (6.0-15.0 p) 


Fig. 6-122. Selective-window transmission [109]. 
















258 


ATMOSPHERIC PHENOMENA 


Table 6-7. Atmospheric Window Definition [109]. 


Window 

Wavelength Limits Window 

Wavelength Limits 

No. 

(n) 

No. 

(n) 

I 

0.72 to 0.94 

VI 

2.70 to 4.30 

II 

0.94 to 1.13 

VII 

4.30 to 6.00 

III 

1.13 to 1.38 

VIII 

6.00 to 15.00 

IV 

1.38 to 1.90 

IX 

15.00 to 25.00 

V 

1.90 to 2.70 



Infrared Radiation Through Clouds 

[110]. 

Clouds become incr 


6.9.I.I. 

transparent to infrared radiation as wavelength is increased. The optical density or 
"thickness” of a cloud is approximately inversely proportional to its visibility. 

Figures 6-123 through 6-127 show cloud attenuation of infrared radiation as a func¬ 
tion of visibility at an altitude of 2500 ft. These data were obtained by viewing an 
infrared source with a detector at various ranges and measuring the change in energy 
viewed by the detector when the measuring path was obscured by clouds of various 
measured densities. The optical density of a cloud was measured by determining the 
maximum range at which a "maximum contrast” object could be discerned in a given 
cloud. Figures 6-123 through 6-127 relate attenuation to "thickness of cloud,” and 
the results do not necessarily apply to the various types of clouds encountered at dif¬ 
ferent altitudes. 



Fig. 6-123. Cloud attenuation vs. cloud visibility, 130 yd measuring range 
(PbS detector) [110]. 



VISIBILITY (yd) VISIBILITY (yd) 


INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


259 



Fig. 6-124. Cloud attenuation vs. cloud visibility, 40 yd measuring 
range (PbS detector) [110]. 



Fig. 6-125. Cloud attenuation vs. cloud visibility, 40 yd measuring 
range (PbSe detector) [110]. 







VISIBILITY (yd) VISIBILITY (yd) 


260 


ATMOSPHERIC PHENOMENA 



Fig. 6-126. Cloud attenuation vs. cloud visibility, 130 yd measuring 
range (PbTe detector) [110]. 



Fig. 6-127. Cloud attenuation vs. cloud visibility, 40 yd measuring 
range (PbTe detector) [110]. 






INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


261 


6.9.2. Slant Paths. Few quantitative data are available on infrared transmission 
over slant paths. The transmission must generally be predicted on the basis of various 
theoretical calculations discussed in this section. 

Both pressure and temperature vary along most slant paths through the atmosphere. 
The half-width of the spectral lines varies with pressure according to Eq. (6-3). Both 
the half-width and intensity of the spectral lines vary with temperature. In addition, 
the concentration of the absorbing gases may vary along a slant path. Because of 
these factors it follows that the equations for slant path absorptance must necessarily 
be more complex than the corresponding homogeneous path equations. 

The mass of absorbing gas per unit area along a slant path is 


u=f Pa 

Jo 


dl 


(6-54) 


where p„ is the density of the absorbing gas 

l is the length of the path measured along the direction of the path. 

Reference [111] describes elaborate calculations for the determination of the amount 
of gas along a slant path. From Fig. 6-128, an equivalent sea-level path can be obtained 
for any gas that is uniformly distributed in the atmosphere, such as C0 2 , N 2 0, CH 4 , 
or CO. The quantity dn is the equivalent path at sea level for a horizontal path 1 km 
long at the indicated altitude. The quantity d v is the equivalent path at sea level for a 
vertical path from the indicated altitude to infinity. 

It is necessary to assume a curve for the variation of H 2 0 concentration with height 
in order to obtain equivalent paths for this gas. In [111], a dry stratosphere with a 
constant mixing ratio in the stratosphere is assumed. There is no agreement at the 
present time between those who favor this distribution and those who favor a wet 
stratosphere (Sec. 6.1.5.2). Figure 6-129 gives equivalent centimeters of precipitable 
water vapor corrected to sea level for a 1-km horizontal path (wh) at the indicated 
altitude and for a vertical path ( w v ) from the indicated altitude to infinity. [Ill] 
The absorptance along a slant path is given by 


A Av 


= L I 1 - ( exp - X *“•<*“■)] 


dv 


(6-55) 


The coefficient is given by Eq. (6-2), (6-4), or (6-6). There are two methods for the 
solution of this problem. In the first method, analytical solutions are obtained for 
Eq. (6-53) under various conditions. In the second method, equivalent path lengths 
and pressures are obtained for a horizontal path so that its absorptance is the same as 
that along the slant path of interest. 

6.9.2.I. Analytical Solutions. When the nonoverlapping line approximation is 
valid (Sec. 6.4.3), an exact expression can be obtained for absorptance over a slant 
path when the absorbing gas is uniformly mixed throughout the atmosphere and the 
temperature variation is small enough so that it can be neglected. The absorptance 
over a slant path at an angle 0 with the vertical is [112] 


A Av = 27raiy 



Pr(z) 



(6-56) 


where z 


1 /Q!o arA , _ Sep p s sec 0 _ Su 

2 W + W 311 y ~ 2tt -get, 27r(<*i-ao) 





262 


ATMOSPHERIC PHENOMENA 



ALTITUDE (km) 

Fig. 6-128. Equivalent sea-level path length for any gas uniformly 
distributed in the atmosphere, e.g., C0 2 , N 2 0, CH 4 , CO [111]. 


a 0 and ai are the half-widths of the spectral lines at the beginning and end of the slant 
path, Co is the fractional concentration of the uniformly mixed absorbing gas, and Py(z) 
is the Legendre function of order y. Equation (6-56) is for a single line, but the ab- 
sorptance values for any number of different lines can be added when they do not over¬ 
lap. The parameter y is similar for slant paths to x for homogeneous paths. The 
absorptance varies linearly with y when y > > 1 and as y 1/2 when y > > 1. 

When y is an integer, Eq. (6-56) can be written as a polynomial in z. For example, 
when y = 1, the absorptance is 


A Av — TTOt\ 



(6-57) 


An exact expression for the absorptance of an Elsasser band can be obtained [112] 
if the same assumptions are made as for the derivation of Eq. (6-56). 




INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


263 



Altitude (km) 


Fig. 6-129. Equivalent centimeters of precipitable water vapor [111]. 


The absorptance by a statistical band (Sec. 6.3.2) is still given by Eq. (6-21) or (6-22) 
when A s i,d is interpreted to be the slant-path absorptance of a single spectral line over 
the frequency interval D. Thus, the absorptance by a statistical band can be obtained 
immediately when any expression for the absorptance of a single line is derived. The 
previous expressions for the random Elsasser (Sec. 6.3.3) and quasirandom (Sec. 6.3.4) 
models can be generalized in similar ways. 

The changing mixing ratio of water vapor over long slant paths must be taken into 
account. It is still possible to obtain analytic expressions for the absorptance which 
converge rapidly in most cases of practical interest. This case is discussed in detail 
in [113] and [114]. 

6.9.2.2. Correspondence Between Slant and Homogeneous Paths. When the weak- 
line approximation is valid, a homogeneous path with an amount of absorbing gas 
Uh and at a temperature such that the individual line intensities are Sm, has the same 
value of the absorptance as a slant path when [113, 114] 






264 


ATMOSPHERIC PHENOMENA 


N N rU 

Uh ^ Si„ = 2 I Si (u) du (6-58) 

The integral in this equation is taken along the slant path with the amount of absorbing 
gas u taken as the independent variable. In general, the line intensity S, varies along 
the slant path because the temperature varies along the path. This result is valid for 
any line shape regardless of whether the line is pressure broadened. It is also valid 
for any distribution of spectral line intensities and for any variation in the spacing 
between adjacent spectral lines. The temperature and absorber concentration may 
vary in any prescribed manner along the slant path. The pressure does not need to 
be specified as long as the weak-line approximation (Sec. 6.4.1) is valid. 

When the temperature variation of the line intensities S, can be neglected along the 
slant path, Eq. (6-58) reduces to the very simple form 


un = u (6-59) 

When the strong-line approximation (Sec. 6.4.2) is valid, and when the temperature 
variation of the line intensities must be taken into account, a useful result can only be 
obtained when the temperature variation of all the spectral lines in the interval can be 
represented by a single function, s(T), so that 


Si = Sih g (T) (6-60) 

Since the temperature variations in the atmosphere are not too large, this equation can 
usually be satisfied with sufficient accuracy. Then, the absorptance along a slant path 
is the same as that for a homogeneous path whose pressure, ph, path length, u/,, and 
temperature T h , satisfy [113, 114] 

p h UhTh~ 112 = J* s(T)p T ~ 1/2 du (6-61) 

The only assumptions made in the derivation of this equation are that the strong-line 
approximation is valid, that there is a pressure-broadened half-width, and that the 
temperature variation of the line intensities can be represented by Eq. (6-60). Equa¬ 
tion (6-61) is valid for any variation of the line intensities, half-widths, and spacing 
between the lines within the spectral interval. 

If, in addition, the temperature variation along the slant path can be neglected and 
this temperature is the same as that along the homogeneous path, then s(T) = 1 and Eq. 
(6-61) simplifies to [113, 114] 


Phu h = J^ pdu (6-62) 


If the absorbing gas is also distributed uniformly along the slant path, and if there 
is the same amount of absorbing gas along the homogeneous and slant paths, Eq. (6-62) 
reduces to [113, 114] 


Ph = 2 (P<> + Pi) 


(6-63) 


where p 0 and pi are the values of the pressure at the two ends of the slant path. 


INFRARED TRANSMISSION THROUGH THE ATMOSPHERE 


265 


If the variation with height of the fractional concentration of a gas such as water 
vapor can be written as some power of the pressure 



(6-64) 


where Co is the fractional concentration when the pressure is po and l is any number, 
then [113] 



(6-65) 


where g is the acceleration of gravity. 

This method of obtaining the absorption along a slant path from homogeneous path 


data is discussed in detail in [113, 114]. Reference [115] gives slant-path absorption 
from laboratory data, using the equations as a basis for calculations. Appropriate 
equivalent sea-level paths for various slant paths were first calculated, and then 
applied to laboratory absorption measurements. 

An important approximate expression, which is valid over a wide range of pressure 
and path length, can be obtained for slant-path absorption by combining the weak-line 
and strong-line methods of determining slant-path absorption from homogeneous path 
data. When the weak-line approximation is valid, a value of Uh is determined from 
Eq. (6-58), which determines the appropriate path length for the homogeneous path. 
On the other hand, a value of the product p u Uh is determined from Eq. (6-61), (6-62), 
(6-63), and (6-65), when the strong-line approximation is valid. 

It is possible to satisfy simultaneously any pair of these weak- and strong-line equa¬ 
tions, since the weak-line equation determines the value of Uh and the strong-line 
equation can then be solved for p h . These results may then be substituted into any 
equation which has been derived, or used in conjunction with any table of measured or 
calculated values of absorptance along a homogeneous path. The slant-path absorp- 
tance calculated in this manner necessarily agrees with the correct result in both the 
weak- and strong-line limits. In the intermediate region where both the weak- and 
strong-line approximations may be somewhat in error, the result derived in this manner 
provides a smooth interpolation between the absorptance curves that are valid in the 
weak- and strong-line limits. In general, the value obtained from this interpolation 
method is very close to the actual value for the slant-path absorptance. 

As an example of this method, when Eq. (6-58) and (6-61) are combined it is found that 



( 6 - 66 ) 


Uh — n 



and 



Pk = 


(6-67) 





266 


ATMOSPHERIC PHENOMENA 


When these values are substituted into any expression for the absorptance along a 
homogeneous path, the resulting equation for the absorptance is necessarily correct 
for a slant path in both the weak- and strong-line limits. 

A more complete discussion of possible combinations of weak- and strong-line expres¬ 
sions and their use with various equations for the absorptance along a homogeneous 
path is given in [113]. This method is a generalization of the interpolation procedure 
described in [116] and [117]. 

A number of calculations of slant-path absorptances over various regions of the spec¬ 
trum have been made. Those first described in [115] are discussed earlier in this 
section. 

Reference [118] gives the transmissivity along a vertical path from a given altitude 
to the top of the atmosphere. The data are based on the equivalent sea-level paths 
given in reference [115]. The calculations apply only to C0 2 and H 2 0 absorption from 
2 to 5 /Lt. 

Reference [111] contains graphs of equivalent sea-level paths for the various ab¬ 
sorbing gases, vapors, and haze which may be encountered along an atmospheric path. 
The absorptivity calculations are based on laboratory data and assume the 1959 ARDC 
model atmosphere. However, methods for correcting the results for various other 
conditions are given. The absorptivity of all important gases is considered from 1 to 
20 ix and the H 2 0 data is extended out to 4 mm. 

Reference [119] gives slant-path absorptivities for H 2 0 and C0 2 from 1 to 10.8 fi. 
Tables are given for computing the C0 2 and H 2 0 amounts along the line of sight and the 
equivalent pressures to be used for the absorptivity calculations. Attempts are made 
to fit each wavelength interval by the absorption model which represents best the data 
for that interval. Reference [120], from the transmission models for H 2 0 and C0 2 
given in [119], presents simplified methods for determining equivalent paths. The 
effect of a curved earth is taken into account. The results are expressed in terms of 
three constants for C0 2 and one constant for H 2 0, which are tabulated at various 
wavelengths. 

A study of the absorption in the 2.7 -\x region of H 2 0 and C0 2 , with simplified band 
models, is described in [121]. An empirical expression to fit the Elsasser band model 
to the 4.3 -fx CO 2 band is discussed in [122]. The slant-path transmission is calculated 
by dividing the atmosphere into small layers and applying Lambert’s law to each layer 
starting from the top of the atmosphere. 

A detailed calculation of the transmissivity from the top of the atmosphere down to 
certain altitudes and at a number of angles to the vertical is given in [123]. Equations 
similar to (6-66) and (6-67) have been used to correlate the appropriate quantities for 
slant and homogeneous paths. The calculations are based on the transmissivity tables 
described in [56] and [57]. The transmissivity is given for each 5-cm _1 interval from 
500 to 10,000 cm -1 for C0 2 and from 1000 to 10,000 cm -1 for H 2 0. 

6.10. Calculation Procedures 

Previous sections of this chapter have discussed in detail the different aspects of the 
earth’s atmosphere as related to atmospheric transmission. In order to determine the 
transmissivity for a specific slant path one must first determine the equivalent sea-level 
path for the absorber of interest. The second step is to find an absorption coefficient 
which is characteristic of the wavelength or the wavelength interval under considera¬ 
tion. This can be carried out by using the figures presented in this chapter or by meas¬ 
ured coefficients given by Howard, Burch, and Williams [107]. The final step in 
determining path transmissivity is to select the functional relation between transmis¬ 
sivity and the product of the absorption coefficient and the reduced optical path which 
best suits the slant path under consideration. 


CALCULATION PROCEDURES 267 

The most tedious part of this process is the determination of the equivalent sea-level 
path. In general the integral relation 


rX 2 

/P\ n /T\ 

£= 

II 

■i 

•ft 

[pj y 


■r, 


( 6 - 68 ) 


must be evaluated along the atmospheric path for the specific set of atmospheric condi¬ 
tions. Here q{x) is the mixing ratio along the slant path; p is the pressure along the 
slant path; T(x) is the temperature along the slant path; p{x) is the density of air along 
the slant path; m and n refer to the power of the temperature and pressure correction for 
line half-width. For high accuracy this relation is best evaluated by digital computer; 
however, Carpenter and Altschuler have each developed graphical techniques which 
apply with reasonable accuracy for a standard model atmosphere. Carpenter’s proce¬ 
dure will be given here. 

Adopting Carpenter’s notation, h s is the source altitude, ha is the detector altitude, 
x g is the ground range between source and detector, and h 0 is the minimum slant-path 
altitude. It is evident, as shown in Fig. 6-130, that the atmospheric paths can be sep¬ 
arated into two types, A and B. Suppose the slant path were extended indefinitely in 



Fig. 6-130. Optical ray paths. 













268 


ATMOSPHERIC PHENOMENA 


both directions. If it then intersected the earth, it would be of type B; otherwise it 
might be considered to be of type A, which is horizontal. 

STEP 1 

Usually h s , h,i, and x g are the known geometric parameters of the slant path. The 
first step is to determine for a given slant path whether the slant path is of type A or 
type B. Figure 6-131 is used for this purpose, and a trial-and-error procedure should 
be carried out. 

To determine whether a path is of type A (the horizontal class) a series of very long 
horizontal paths has been constructed (Fig. 6-131). Such paths might be referred to as 
fundamental horizontal paths. Any slant path of type A will be a segment of one of the 
illustrated paths. Note that each fundamental path is uniquely specified by its mini¬ 
mum altitude ho. Therefore if one wishes to determine whether a given slant path is of 
type A or B, two horizontal lines would be drawn on Fig. 6-131 with the first at the alti¬ 
tude of the source and the second at the altitude of the receiver. These horizontal lines 
will intersect many of the fundamental paths. If one of the intersected paths has 
approximately the appropriate ground distance, then this fundamental path contains as 
a segment the slant path of interest. From the intersection of the horizontal lines with 
the fundamental path the specific elevation angles at the source point and receiver 
point can be read together with the corresponding value of ho for the path. 

From Fig. 6-130 it can be seen that point Q may or may not be a part of the desired 
slant path. If point Q does not fall on the slant path but the path is of type A, the path- 
determination procedure is as described above. If point Q falls within the slant path 
then it should be realized that Fig. 6-131 is symmetric about the ordinate axis. Under 
this condition the figure might be redrawn to include the unpublished half. The path 
determination would proceed as described. 

Slant paths of type B can be divided into two cases. Case I applies whenever 86° 
> i > 90°; case II applies if 0 > i > 86°. 

Example 1 


Example 2 


Let ha = 15,000 ft 
h s = 30,000 ft 
= 150,000 ft 

type A 

h 0 = 10,000 ft 

h d = 15,000 ft 
h s = 30,000 ft 

Xg = 500,000 ft 

type B, case I 

ho = 10,000 ft 

i — 1.10 

h d = 15,000 ft 
h s = 30,000 ft 

type B , case II 

lo = 0 

V3 ... 

i = 600 


x g = — 30,000 ft 


Example 3 


CALCULATION PROCEDURES 


269 






















270 


ATMOSPHERIC PHENOMENA 


STEP 2 

The second step for carbon dioxide is to determine the equivalent optical path for an 
infinite horizontal slant path, Uh, or a vertical path, u v . 

Use Fig. 6-132 to get ui, for atmospheric paths of type A 
Use Fig. 6-132 to get Uh for slant paths of type B, case I 

Use Fig. 6-133 to get u v for slant paths of type B, case II, where in Fig. 6-133 the 
abscissa is either h d or hg, whichever is smaller. 


io 4 




E 

U 


E 

15 



10 


1 





Fig. 6-132. Total horizontal equivalent optical Fig. 6-133. Scale height for 

path length for C0 2 us. altitude. 


STEP 3 

For Carpenter’s procedure, step 3 is the determination of a scale height H 0 " which 
can be found in Fig. 6-134 for all atmospheric paths involving C0 2 . 

From steps 1 or 2 the value of h 0 has been determined. To find Ho, one simply reads 
the appropriate value from Fig. 6-134. 






CALCULATION PROCEDURES 


271 



Fig. 6-134. Total vertical equivalent optical path length for C0 2 
vs. altitude. 


STEP 4 


To compute the equivalent sea-level paths for atmospheric paths of type A, use 




u e = u h 


erf 


ht-hoY 12 , Ao\ ,/2 


Ho" 


+ erf 


/ h s h 0 \ 
\ Ho' ) 


(6-69) 


For atmospheric paths of type B, case I, use 


U e = Uh 


e. h.-hoV 2 r (hs~ ko\ 1/2 

eTf[ ~Hr) erf m 


(6-70) 


For atmospheric paths of type B, case II, use 




hg hd 


Ue = sec i u v ( h(t ) 

1 — exp — 

L Ho" J 



(6-71) 


































(H 0 0) (g/cm or cm precipitable) 


272 


ATMOSPHERIC PHENOMENA 


STEPS 2-4, for water vapor when the mixing ratio is known. When the mixing ratio 
m is constant for all altitudes, the procedure is identical for both carbon dioxide and 
water vapor with the following exceptions: use Fig. 6-135 instead of Fig. 6-132, use 
Fig. 6-136 instead of Fig. 6-133 and use Fig. 6-135 again for Fig. 6-135. 



Fig. 6-135. Total horizontal equivalent optical 
path length for H 2 0 vs. altitude (for mixing 
ratios 1, 0.1, 0.01). 


Fig. 6-136. Total vertical equivalent optical 
path length for H 2 0 vs. altitude (for mixing 
ratios 1, 0.1, 0.01). 













CALCULATION PROCEDURES 


273 


STEPS 2-4 for constant relative humidity. When a constant relative humidity is 
assumed for the atmospheric path, the procedure is again similar to that given above 
with the exception that Uh should be determined from Fig. 6-137, u v should be de¬ 
termined from Fig. 6-138, and Ho" replaces Ho in Eq. (6-69), (6-70), and (6-71). Use 
Fig. 6-139 for the determination of Ho". 

The reduced equivalent sea-level paths obtained using the above procedure are cor¬ 
rected for temperature and pressure as well as for the refraction of the atmospheric 
path. Values of the reduced sea-level path computed in this way will be quite accurate 
for temperate latitudes and may be used for the computation of slant-path transmissivi¬ 
ties as outlined in the previous sections. 




Fig. 6-137. Total horizontal equivalent optical Fig. 6-138. Total vertical equivalent optical 
path length for H 2 0 vs. altitude (for relative hu- path length for H 2 0 vs. altitude (for relative 
midity 100%, 10%, 1%). humidity 100%, 10%, 1%). 




















22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 


ATMOSPHERIC PHENOMENA 



I 


i 


Water 


I q Ice 

I X 


' I 
o°c 


X 


X 




20 


40 


60 


80 


100 120 


ALTITUDE, h Q (xlO ft) 


140 


Fig. 6-139. Slant height for P 2 IT 3/2 . 

















REFERENCES 


275 


References 

1. A. E. Cole, A. Court, and A. J. Kantor, Standard Atmosphere Revision to 90 Kilometers, Report 
No. INAP-7, Rev. 2, Geophysics Research Directorate, AFCRL, Cambridge, Mass. (1961). 

2. A. E. Cole, A. Court, and A. J. Kantor, Supplemental Atmospheres, Geophysics Research 
Directorate, AFCRL, Cambridge, Mass. (1963). 

3. Handbook of Geophysics-Revised Edition, Macmillan Company, New York (1960). 

4. C. E. Junge, Atmospheric Chemistry, Advances in Geophysics, Vol. 4, Academic Press, New 
York (1958). 

5. M. Gutnick, "How Dry is the Sky?,” J. of Geo. Res., 66, No. 9, 2867 (1961). 

6. J. T. Houghton and J. S. Seeley, "Spectroscopic Observations of the Water Vapor Content of 
the Atmosphere,” Quart. J. Roy. Meteor. Soc., 86, 358 (1960). 

7. J. Yarnell and R. M. Goody, "Infrared Solar Spectroscopy in a High-altitude Aircraft,” J. Sci. 
Instr., 29, 352 (1952). 

8. A. W. Brewer, "Evidence for a World Circulation Provided by the Measurements of Helium 
and Water Vapor Distribution in the Stratosphere,” Quart. J. Roy. Meteor. Soc., 75, 351 (1949). 

9. G. M. B. Dobson, "Origin and Distribution of the Polyatomic Molecules in the Atmosphere,” 
Proc. Roy. Soc., London, 236, No. 1205 187 (1956). 

10. P. Goldsmith, "Some Aircraft and Surface Meteorology Observation Made at Khartoum,” 
Meteorol. Mag., 23, 329 (1954). 

11. N. C. Helliwell and J. K. Mackenzie, Observations of Humidity, Temperature and Wind at 
Idris, 23rd May-2nd June 1956, MRP 1024, Meteorological Research Committee, London 
(1957). 

12. N. C. Helliwell, J. K. Mackenzie, and M. J. Kerley, Further Observations of Humidity Up to 
50,000 Feet, Made from an Aircraft of the Meteorological Research Flight in 1955, MRP 976, 
Meteorological Research Committee, London (1956). 

13. N. C. Helliwell, J. K. Mackenzie and M. J. Kerley, "Some Further Observations from Aircraft 
of Frost Point and Temperature up to 50,000 Feet,” Quart. J. Roy. Meteor. Soc., 83, 257 (1957). 

14. R. J. Murgatroyd, P. Goldsmith, and W. E.H. Hollings,"Some Recent Measurements of Humid¬ 
ity from Aircraft up to Heights of about 50,000 Feet over Southern England,” Quart. J. Roy. 
Meteor. Soc., 81, 533 (1955). 

15. D. M. Gates, D.G. Murcray, C.C. Shaw, and R. J. Herbold, "Near Infrared Solar Measurements 
by Balloons to Altitudes of 100,000 Feet,” J. Opt. Soc. Am., 48, 1010 (1958). 

16. D. G. Murcray, F. H. Murcray, W. J. Williams, and F. E. Leslie, Water Vapor Distribution 
Above 90,000 Feet, Scientific Report No. 5, Geophysics Research Directorate, Contract AF 19 
(604)-2069, AFCRL, Cambridge, Mass. (1960). 

17. N. Sissenwine and M. Gutnick, "Precipitable Water Along High Altitude Ray Paths,” Proc. 
of Infrared Information Symposium, 5, No. 2, 5 (1960). 

18. H. J. Mastenbrook, and J. E. Dinger, The Measurement of Water Vapor Distribution in the 
Stratosphere, NRL Report 5551, U.S. Naval Research Laboratory, Wash., D.C., (1960) (ASTIA 
AD 247760). 

19. M. J. Mastenbrook and J. E. Dinger, "Distribution of Water Vapor in the Stratosphere,” 
J. Geophysics, 66, 1437 (1961). 

20. F. R. Barclay, M. J. Eliott, P. Goldsmith, and J. V. Jelley, "A Direct Measurement of the 
Humidity in the Stratosphere Using a Cooled-Vapor Trap,” Quart. J. Roy. Meteor. Soc. 86, 
259 (1960). 

21. F. Stauffer and J. Strong, App. Optics, 1, (1962). 

22. J. Strong, Preliminary Report on Solar Observations from a U-2 Observatory, The Johns 
Hopkins University, Baltimore, Md., (1960). 

23. D. G. Murcray, F. H. Murcray, and W. J. Williams, Distribution of Water Vapor in the Strato¬ 
sphere as Determined from Infrared Absorption Measurements, Scientific Report No. 1, Uni¬ 
versity of Denver, Contract No. AF 19 (604)-7429, AFCRL Report No. 219, AFCRL, Cam¬ 
bridge, Mass., (1961). 

24. R. K. McDonald, Infrared Satellite Background Measurements, Part 1, Atmospheric Radiative 
Processes, Final Report, Sept. 1961, The Boeing Company, Contract AF 19 (604)-7457 for 
Geophysics Research Directorate, AFCRL Report No. 1069, AFCRL, Cambridge, Mass. 

25. T. L. Altshuler, Infrared Transmission and Background Radiation by Clear Atmospheres, 
Document No. 615D199, Dec. 1961, General Electric Company, Missile and Space Vehicle 
Department, Philadelphia, Pa. 

26. M. Gutnick, Mean Moisture Profiles to 31 Kilometers for Middle Latitudes, Geophysics Re¬ 
search Directorate, AFCRL, Cambridge, Mass., (1962). 

27. W. L. Godson, "Total Ozone and the Middle Stratosphere Over Arctic and Subarctic Areas in 
Winter and Spring,” Quart. J. Roy. Meteor. Soc., 86, No. 369, 301 (1960). 


276 


ATMOSPHERIC PHENOMENA 


28. E. O. Hurlburt, "Physics of the Upper Atmosphere,” Meteor. Res. Rev. 3, No. 17, 167 (1957). 

29. R. M. Goody and W. T. Roach, "Determinations of the Vertical Distribution of Ozone from 
Emission Spectra,” Quart. J. Roy. Meteor. Soc., 82, No. 352, 217 (1956). 

30. H. Johansen, On the Relation Between Meteorological Conditions and Total Amount of Ozone 
Over Tromso, Polar Atmosphere Symposium, Part I, Pergamon Press, London, 187 (1958). 

31. C. L. Mateer and W. L. Godson, "The Vertical Distribution of Atmospheric Ozone Over 
Canadian Stations from Umkehr Observations,” Quart. J. Roy. Meteor. Soc., 18, No. 3, 512 
(1960). 

32. H. U. Dutsch, Current Problems of the Photochemical Theory of Atmospheric Ozone, paper 
given at Inter. Symp. on Chem. Reac. in the Lower and Upper Atmosphere, Stanford Research 
Institute, Stanford, California, April 18-20, 1961. 

33. K. R. Ramanthan and R. N. Kulkarni, "Mean Meridional Distribution of Ozone in Different 
Seasons,” Calculated from Umkehr Observations and Probable Vertical Transport Mech¬ 
anisms, Quart. J. Roy. Meteor. Soc., 86, No. 368, 144 (1960). 

34. H. K. Paetzold, The Photochemistry of the Atmospheric Ozone-Layer, paper given at Inter. 
Symp. on Chem. Reac. in the Lower and Upper Atmosphere, Stanford Research Institute, 
Stanford, California, April 18-20 (1960). 

35. S. Verkateswaren, J. G. Moore, and A. J. Krueger, "Determination of the Vertical Distribution 
of Ozone by Satellite Photometry,” J. Geophys. Res., 66, 1751, University of Calif., Berkeley, 
Calif., Naval Ordnance Test Station (1961). 

36. J. N. Howard and J. S. Garing, The Transmission of the Atmosphere in the Infrared, GRD, 
AFCRL, Cambridge, Mass. (1962). 

37. C. C. Junge, "Atmospheric Composition,” Handbook of Geophysics for Air Force Designers, 
AFCRC, Cambridge, Mass. (1957). 

38. A. J. Arnulf, J. Bricard, E. Cure and C. Veret, "Transmissions by Haze and Fog in the Spectral 
Region 0.35 to 10 Microns,” J. Opt. Soc. Am., 47, 491 (1957). 

39. H. J. Weickman and J. J. aufm Kampe, "Physical Properties of Cumulus Clouds,” J. Meterol., 
10, No. 204 (1953). 

40. D. Deirmendjian, Scattering and Polarization Properties of Polydispersed Suspension with 
Partial Absorption, Rand Corp., Memorandum RM 3228 PR. 

41. R. Penndorf, The Vertical Distribution of Mie Particles in the Troposphere, Geophysical 
Research Paper No. 25, AFCRC Tech. Report 54-5, AFCRC, Cambridge, Mass. (1954). 

42. D. E. Burch, E. B. Singleton, and D. Williams, App. Optics, 1, No. 359 (1962). 

43. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley 
Publishing Co., Inc., Reading, Mass. (1959). 

44. G. N. Plass and D. I. Fivel, Astrophys. J., 117, 225 (1953). 

45. W. M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere, Harvard Meteor 
Studies No. 6, Harvard University Press, Cambridge, Mass. (1942). 

46. G. N. Plass, J. Opt. Soc. Am., 48, 690 (1958). 

47. G. N. Plass, J. Opt. Soc. Am., 50, 868 (1960). 

48. R. Ladenberg and F. Reiche, Ann. Physik., 42, 181 (1913). 

49. G. N. Plass and D. I. Fivel, Astrophys. J., 117, 225 (1953). 

50. W. M. Elsasser, Phys. Rev., 54, 126 (1938). 

51. H. Mayer, Methods of Opacity Calculations, Los Alamos, LA-647 (1947). 

52. R. M. Goody, Quart. J. Roy. Meteor. Soc., 78, 165 (1952). 

53. L. D. Kaplan, Proc. 1953 Toronto Meteor. Conf., 43 (1954). 

54. P. J. Wyatt, V. R. Stull, and G. N. Plass, J. Opt. Soc. Am. (1962). 

55. D. Q. Wark and M. Wolk, Monthly Weather Review, 88, 249 (1960). 

56. P. J. Wyatt, V. R. Stull, and G. N. Plass, App. Optics, 3 (1964); Aeronutronic Report U-1717, 
Aeronutronic Systems, Inc., Newport Beach, Calif. (1962). 

57. V. R. Stull, P. J. Wyatt, and G. N. Plass, App. Optics, 3 (1964); Aeronutronic Report U-1718, 
Aeronutronic Systems, Inc., Newport Beach, Calif., 1962. 

58. W. E. K. Middleton, Vision Through the Atmosphere, University of Toronto Press, Toronto, 
Canada (1952) Section 9.3.1.1. 

59. W. E. K. Middleton, "The Effect of the Angular Aperture of a Telephotometer on the Tele¬ 
photometry of Collimated and Non-Collimated Beams,” J. Opt. Soc. Am., 39, No. 576 (1949). 

60. H. S. Stewart and J. A. Curcio, "The Influence of Field-of-View on Measurements of Atmos- 
spheric Transmission,” J. Opt. Soc. Am., 42, No. 801 (1952). 

61. J. W. Tucker, Computation of Singly-Scattered Radiation from a Distaat Source when the 
Angular Scattering Function is Known, NRL Report 5260, U.S. Naval Research Laboratory, 
Wash., D.C. (1959). 

62. K. Bulrich, "Die Streuung des Lichter in Truber Luft,” Optik, 2, No. 301 (1947). 

63. R. Gans, "Wien-Harms Handbuch D. Exp.,” Physik, 19, No. 368. 


REFERENCES 277 

64. J. A. Stratton and H. G. Houghton, "A Theoretical Investigation of the Transmission of 
Light Through Fog,” Phys. Rev., 38, No. 159 (1931). 

65. J. S. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941). 

66. H. C. Van De Hulst, Light Scattering by Small Particles, Wiley, New York (1957). 

67. H. G. Houghton and W. R. Cholker, "The Scattering Cross-Section of Water Crops in Air 
for Visible Light,” J. Opt. Soc. Am., 39, No. 955 (1949). 

68. Tables of Scattering Functions for Spherical Particles, NBS Applied Math., Series 4, National 
Bureau of Standards, Wash., D.C. (1949). 

69. C-N Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for 
Light Scattering by Spheres, Engineering Research Institute, Department of Chemical and 
Metallurgical Engineering, The University of Michigan Press, Ann Arbor, Mich. (1957). 

70. J. Bricard, "Etude de la constitution des nuages au sommet du Puy-de-Dome,” La Meterologie, 
15, 83. 

71. J. Bricard, "Lumiere diffusee en avante par une gotte d’eau spherique,” J. de Phys. et la 
Radium, 4, 57. 

72. J. Bricard, "Reflexion, refraction et diffraction de la lumiere par une gotte d’eau spherique,” 
Ann. de Geophys., 2, 231. 

73. P. Kruse, L. McGlaughlin, and R. McQuistan, Elements of Infrared Technology, Wiley, New 
York (1962). 

74. H. W. Yates and J. H. Taylor, Infrared Transmission of the Atmosphere, NRL Report 5453, 
U.S. Naval Research Laboratory, Wash., D.C. (1960). 

75. M. G. Gibbons, "Transmission and Scattering Properties of a Nevada Desert Atmosphere,” 
J. Opt. Soc. Am., 51, No. 633 (1961). 

76. J. A. Curcio, G. L. Knestrick, and T. H. Cosden, Atmospheric Scattering in the Visible and 
Infrared, NRL Report 5567, U.S. Naval Research Laboratory, Wash., D.C. (1961) ASTIA 
AD 250945. 

77. L. P. Granath and E. 0. Hulburt, "The Absorption of Light by Fog,” Phys. Rev., 34, No. 140 
(1929). 

78. J. A. Curcio and K. A. Durbin, Atmospheric Transmission in the Visible Region, NRL Report 
5368, U.S. Naval Research Laboratory, Wash., D.C. 

79. F. Lohle, "Uber die Lichtzerstreuung im Nebel,” Phys. Zeits., 45, No. 199 (1944). 

80. M. Wolff, "Die Lichttechnischen Eigenschaften des Nebels,” Das Licht, 8, No. 105 and 128 
(1938). 

81. Measurement of Atmospheric Attenuation aboard USAS AMERICAN MARINER, Project 
DAMP Progress Report, Barnes Eng. Co., Stamford, Conn. 

82. H. B. Glenn, Light Transmission Through an Apparently Clear Atmosphere, Raytheon Co., 
Santa Barbara Research Operations, Santa Barbara, Calif. (1960). 

83. W. M. Protheroe, Preliminary Report on Stellar Scintillation, Scientific Report No. 4, Ohio 
State University, Contract No. AF 19 (604)-41, AFCRL, Cambridge, Mass., (1954) ASTIA 
AD 56040. 

84. G. Keller, W. M. Protheroe, P. E. Barnhard and J. Galli, Investigation of Stellar Scintillation 
and the Behavior of Telescopic Images, Final Report, Contract No. AF 19 (604)-1409, AFCRC, 
Report No. TR-57-186, AFCRC, Cambridge, Mass. (1956) ASTIA AD 117279. 

85. E. Goldstein, The Measurement of Fluctuating Radiation Components in the Sky and Atmos- 
sphere, Part 1, NRL Report N-3462, U.S. Naval Research Laboratory, Wash., D.C. (1949) 
ASTIA ATI-7149. 

86. E. Goldstein, The Measurement of Fluctuating Radiation Components in the Sky and Atmos- 
sphere, Part 2, NRL Report 3710, U.S. Naval Research Laboratory, Wash., D.C. (1950) ASTIA 
PB-102617. 

87. F. R. Bellaire and F. C. Elder, Scintillation and Visual Resolution Over the Ground, Report 
of Project MICHIGAN, 2900-134-T, The University of Michigan, Willow Run Lab., Ann Arbor, 
Mich. (1960) ASTIA AD 245118. 

88. B. N. Bullock, G. M. Smith, and R. P. Borofka, Atmospheric Boil Measurements, ITT Federal 
Laboratories, San Fernando, Calif. 

89. F. Benford, "Duration of Intensity of Sunshine - Part I - General Considerations and Correc¬ 
tions,” Illuminating Engineering, 42, 527, General Electric Co. (1947). 

90. M. Migeotte, L. Nevin, and J. Swensson, The Solar Spectrum from 2.8 to 23.7 Microns, Part I. 
Photometric Atlas, University of Liege, Contract AF 61 (514)-432, Phase A, Part I, Geophysics 
Research Directorate, AFCRC, Cambridge, Mass. ASTIA AD 210043. 

91. M. Migeotte, L. Nevin, and J. Swensson, The Solar Spectrum from 2.8 to 23.7 Microns, Part II. 
Measures and Identifications, University of Liege, Contract AF 61(514)-432, Phase A, Part II, 
Geophysics Research Directorate, AFCRC, Cambridge, Mass. ASTIA AD 210044. 


ATMOSPHERIC PHENOMENA 


278 

92. M. Migeotte, L. Nevin, and J. Swensson, An Atlas of Nitrous Oxide, Methane and Ozone 
Infrared Absorption Bands, Part I. The Photometric Records, University of Liege, Contract 
AF 61(614)-432, Phase B, Part I, Geophysics Research Directorate, AFCRC, Cambridge, 
Mass. ASTI A AD 210045. 

93. M. Migeotte, L. Nevin, and J. Swensson, An Atlas of Nitrous Oxide, Methane and Ozone 
Infrared Absorption Bands, Part II. Measures and Identifications, University of Liege, 
Contract AF 61(514)-432, Phase B, Part II, Geophysics Research Directorate, AFCRC, Cam¬ 
bridge, Mass. ASTIA AD 210046. 

94. J. N. Howard and J. S. Garing, Infrared Atmospheric Transmission: Some Source Papers on 
the Solar Spectrum from 3 to 15 Microns, Air Force Surveys in Geophysics No. 142, AFCRL 
Report No. 1098, Dec. 1961, Geophysics Research Directorate, AFCRL, Cambridge, Mass. 

95. J. N. Howard, "Atmospheric Transmission in the 3 to 5 Micron Region,” Proc. IRIS, 2, 59-75 
(1957). 

96. J. N. Howard, "Atmospheric Transmission in the 8 to 13 Micron Region,” Proc. of the Sympo¬ 
sium on Optical Radiation from Military Airborne Targets, Final Report No. AFCRL-TR-58- 
146, AFCRL, Cambridge, Mass., Contract No. AF 19(604)-2451, Haller, Raymond and Brown, 
Inc., State College, Pa. ASTIA AD 152411; see also [7]. 

97. O. C. Mohler, A. K. Pierce, P. R. McMath, and L. Goldberg, Atlas of the Solar Spectrum from 
0.84 to 2.52 Microns, University of Michigan Press, Ann Arbor, Mich. (1950). 

98. O. C. Mohler, Table of Solar Spectrum Wavelengths from 1.20 to 2.55 Microns, University of 
Michigan Press, Ann Arbor, Michigan (1955). 

99. J. H. Shaw, R. M. Chapman, J. N. Howard, and M. L. Oxholm, "A Grating Map of the Solar 
Spectrum from 3.0 to 5.0 Microns,” Astrophys. J., 113, No. 2, (1951) see also [7]. 

100. J. H. Shaw, M. L. Oxholm, and H. H. Classen, "The Solar Spectrum from 7 to 13 Microns,” 
Astrophys. J., 116, No. 3 (1952) see also [7]. 

101. W. W. Talbert, H. A. Templin, and R. E. Morrison, Quantitative Solar Spectral Measurements 
at Mt. Chacaltaya ( 17,100 ft.), U.S. Naval Ordnance Laboratory, White Oak, Maryland; See 
also J. Opt. Soc. Am., 47, 156 (1957). 

102. J. E. Seeley, J. T. Houghton, T. S. Moss, and N. D. Hughes, "Solar Spectrum from 1 to 6.5 
Microns at Altitudes Up to 15 KM,” Phil. Trans. Roy. Society. 

103. C. B. Farmer and S. J. Todd, Reduced Solar Spectra 3.5 to 5.5 Microns, Report DP 927 E.M.I. 
Electronics, Hayes, Middlesex, England (1961). 

104. D. E. Burch and D. Williams, Infrared Absorption by Minor Atmospheric Constituents, The 
Ohio State University Research Foundation, Scientific Report No. 1, Contract No, AF 19(604)- 
2633, Geophysics Research Directorate, AFCRL Report No. TN-60-674, AFCRL, Cambridge, 
Mass. (1960) ASTIA AD 246921. 

105. D. E. Burch, D. Gryvnak, and D. Williams, Infrared Absorption by Carbon Dioxide, The Ohio 
State University Research Foundation, Scientific Report No. 11, Contract No. AF 19(604)- 
2632, Geophysics Research Directorate, AFCRL Report No. 255, AFCRL, Cambridge, Mass. 
(1960) ASTIA AD 253435. 

106. D. E. Burch, E. B. Singleton, W. L. France, and D. Williams, Infrared Absorption by Minor 
Atmospheric Constituents, The Ohio State University Research Foundation, Final Report, 
Contract No. AF 19(604)-2633. Geophysics Research Directorate, AFCRL Report No. 412, 
AFCRL, Cambridge, Mass. (1960) ASTIA AD 256952. 

107. J. N. Howard, D. E. Burch, and D. Williams, Near-Infrared Transmission Through Synthetic 
Atmospheres, AFCRL Report No. TR-55-213, Geophysics Research Directorate, AFCRL, 
Cambridge, Mass. (1955). 

108. H. W. Yates and J. H. Taylor, Infrared Transmission of the Atmosphere, NRL Report 5453, 
U.S. Naval Research Laboratory, Wash., D.C. (1960) ASTIA AD 240188. 

109. T. Elder and J. Strong, "The Infrared Transmission of the Atmospheric Windows,” Journal of 
the Franklin Institute, 255, No. 3, 189 Phila., Pa. (1953). 

110. C. B. Farmer, The Transmission of Infrared Radiation Through Cloud, Report No. DP. 841, 
E.M.I. Electronics, Ltd., Hayes, Middlesex, England, (1960). 

111. T. L. Altshuler, Infrared Transmission and Background Radiation by Clear Atmospheres, 
General Electric Company, Missile and Space Vehicle Department, Valley Forge, Pa., Docu- 
61SD199 (1961). 

112. G. N. Plass, and D. I. Fivel, J. Meteorol., 12, 191 (1955). 

113. G. N. Plass, App. Optics 2, 515 (1963). 

114. G. N. Plass, J. Opt. Soc. Am., 42, 677 (1952). 

115. R. O’B. Carpenter, J. A. Wight, A. Quesda, and R. E. Swing, Predicting Infrared Molecular 
Attenuation for Long Slant Paths in the Upper Atmosphere, AFCRC Report No. TN-58-253, 
AFCRC, Cambridge, Mass. (1957). 

116. A. R. Curtis, Quart. J. Roy. Meteor. Soc., 78, 638 (1952). 


REFERENCES 


279 


117. W. L. Godson, J. Meteorol., 12, 272 (1955). 

118. A. Thomson and M. Downing, Earth Background and Atmospheric Transmission in the 2 to 5 
Micron Region, Convair Report Ph-069-M (1960). 

119. A. S. Zachor, Near Infrared Transmission Over Atmospheric Slant Paths, Report R-328, 2, 
Massachusetts Institute of Technology, Cambridge, Mass., Contract AF 33(616)-6046 (1961). 

120. A. E. S. Green and M. Griggs, Appl. Optics 2, 561 (1963). 

121. G. A. Morton and G. M. Weyl, Water Vapor and Carbon Dioxide Absorption in the Spectral 
Region Around 2.7 Microns, Special Report No. 2075, Aerojet-General Corp., Azusa, Calif. 
(1961). 

122. L. R. Megill and P. M. Jamnick, J. Opt. Soc. Am., 51, No. 1294 (1961). 

123. G. N. Plass, Transmittance of Carbon Dioxide and Water Vapor over Stratospheric Slant Paths, 
Aeronutronic Report, Aeronutronic Systems, Inc., Newport Beach Calif. (1962); Appl. Optics, 
3 (1964). 




































. 
















































■ 
































Chapter 7 

OPTICAL COMPONENTS 

William L. Wolfe 

The University of Michigan 


CONTENTS 


7.1. Lenses. 282 

7.2. Mirrors. 285 

7.3. Catadioptric Systems. 285 

7.4. Interference Filters. 286 

7.4.1. Introduction. 286 

7.4.2. Terminology. 286 

7.4.3. General Theory of Interference Filters. 288 

7.4.4. Long-Wave Pass Interference Filters. 290 

7.4.5. Short-Wave Pass Interference Filters. 290 

7.4.6. Bandpass Interference Filters. 290 

7.4.7. "Square-Band” Interference Filters. 291 

7.4.8. The Filter Matrix. 291 

7.4.9. The Herpin Equivalent Layer. 293 

7.4.10. Analogies with Transmission-Line Theory. 293 

7.4.11. Effects of Angle of Incidence. 293 

7.4.12. Effects of Temperature. 294 

7.4.13. Substrates and Films. 294 

7.5. Christiansen Filters. 295 

7.6. Selective Reflection Filters. 297 

7.7. Selective Refraction Filters. 298 

7.8. Polarization Interference Filters. 298 

7.9. Commercially Available Filters. 299 

7.10. Absorption Filters. 306 

7.11. Prisms. 307 

7.11.1. Dispersing Prisms. 308 

7.11.2. Deviating Prisms. 308 

7.11.3. Prism Materials. 309 

7.12. Diffraction Gratings. 309 

7.12.1. Blazed Gratings. 310 

7.12.2. Concave Gratings. 311 

7.12.3. Ebert-Fastie Plane Grating Mountings. 311 

7.12.4. Concave Grating Mountings. 311 

7.12.5. Production of Gratings. 313 


281 





































7. Optical Components 


7.1. Lenses 

Infrared lenses and lens systems are usually custom made and are often designed 
by manufacturers. Stock items which are available are made from the more rugged 
materials such as Irtran, arsenic trisulfide, silicon, germanium, and optical glass; 
these materials are described in Chapter 8. Optical glass lenses are not discussed 
because they are easily obtained in almost any aperture and focal length.* 

Tables 7-1 through 7-3 and Figs. 7-1 through 7-5 show various types of lenses and 
lens systems supplied by manufacturers for infrared applications. 

The theory of refracting optics including thin and thick lenses and multiple lens 
systems is covered in detail in Chapter 9. 


Table 7-1. Single-Element Lenses Supplied by Eastman Kodak Company [1]. 
(All are Irtran-2, Meniscus, measurements are in inches) 


Range 

AX 

Nominal Focal 
Length 

f 

Equivalent 
Focal Length 
e.f.1 

Back Focal 
Length 
b.f.l. 

Focal Ratio 
flno 

Circle of 
Confusion 
dt 

Distance 

C. ofC. 
from Vertex 

Lens 

No. 

1.5-10 

1 

0.99* 

0.90* 

1 

0.028 

0.84 

IR-100 



1.04 

0.92 


0.028 

0.84 


1.5-10 

2 

1.99 

1.80 

1 

0.060 

1.66 

IR-200 



2.12 

1.91 


0.055 

1.77 


3-10 

1 

1.00 

0.86 

1 

0 

0.86 

IR-101 



1.05 

0.90 


0.011 

0.87 


4.26-10 

2 

1.92 

1.68 

1 

0.001 

1.68 

IR-201 



2.00 

1.75 


0.001 

1.75 


4.26-10 

3 

3.00 

2.72 

1 

0.001 

2.72 

IR-301 



3.13 

2.84 


0.001 

2.84 



*The first number is for the short-wavelength end of the range; the second is for the long wavelength end. 
tTo the 5% intensity point; measurements made 2.0 to 4.5 /x. 




All Lenses Germanium 

Fig. 7-2. SCA doublet [2]. 


*For extensive lists see the catalogs of appropriate companies listed in the Optical Industry 
Directory. 


282 







































LENSES 


283 


Table 7-2. Single-Element Infrared Lenses Supplied by 
Servo Corporation of America [2]. 


Lens 

Shape 

Range 

AX 

ip) 

Design X 
Xd 

(/a) 

Nominal 

Focal 

Length 

f 

(in.) 

Effective 
Focal Length 
at X d (in.) 
e.f.l. 

Back 

Focal Length 
at Xd (in.) 
b.f.l. 

fl no 
(in.) 

Circle of 
Confusion 
dia. at Xd 
(in.) 

Circle of 
Confusion 
over AX 
(in.) 

Material 


0.7-1.5 

1.1 

4.0 

4.0 

3.683 

4 

0.006 

0.011 

Fused quartz 




2.0 

2.0 

1.753 

2 

0.024 

0.023 

Fused quartz 




4.0 

4.0 

3.523 

2 

0.047 

0.055 

Fused quartz 


0.7-2.0 

1.4 

2.0 

2.0 

1.748 

2 

0.027 

0.030 

CaF 2 

Convex- 



4.0 

4.0 

3.491 

2 

0.053 

0.060 

CaF 2 

Convex 



4.0 

4.0 

3.665 

4 

0.007 

0.011 

CaF 2 


0.7-3.0 

1.8 

2.0 

2.0 

1.759 

2 

0.022 

0.028 

BaF 2 




4.0 

4.0 

3.713 

4 

0.006 

0.011 

BaF 2 




4.0 

4.0 

3.536 

2 

0.044 

0.055 

BaF 2 


2.0-5.0 

5.0 

3.0 

_ 

_ 

0.8 

_ 

_ 

Servofrax 

Plano- 


5.0 

2.0 

— 

— 

2.0 

— 

— 

Servofrax 

Convex 


5.0 

0.75 

— 

— 

0.8 

— 

— 

Servofrax 



4.0 

0.57 

- 

- 

0.65 

- 

- 

Servofrax 


2.0-5.0 

4.0 

0.55 

_ 

_ 

0.63 

_ 

_ 

Servofrax 

Equi- 


6.0 

2.0 

— 

— 

1.0 

— 

— 

Servofrax 

convex 


6.0 

4.0 

- 

- 

1.3 

- 

- 

Servofrax 


1.0-2.0 

1.5 

2.0 

2.0 

1.853 

2 

0.018 

0.022 

MgO 




4.0 

4.0 

3.877 

4 

0.005 

0.003 

MgO 


2.0-5.0 

3.5 

0.75 

0.75 

0.647 

1 

0.024 

0.027 

Servofrax 




1.5 

1.5 

1.378 

3 

0.003 

0.005 

Servofrax 




2.0 

2.0 

1.905 

4 

0.0006 

0.0025 

Servofrax 




2.0 

2.0 

1.905 

2 

0.005 

0.009 

Servofrax 




3.6 

3.59 

3.464 

1.2 

0.044 

0.051 

Servofrax 




4.0 

4.0 

3.835 

4 

0.001 

0.005 

Servofrax 




4.0 

4.0 

3.835 

2 

0.010 

0.017 

Servofrax 




5.5 

5.5 

5.094 

1 

0.176 

0.190 

Servofrax 




8.0 

8.0 

7.671 

4 

0.0025 

0.010 

Servofrax 




8.0 

8.0 

7.671 

2 

0.021 

0.037 

Servofrax 




9.0 

9.0 

8.796 

3 

0.019 

0.030 

Servofrax 

Meniscus 



14.3 

14.29 

13.889 

1.9 

0.041 

0.066 

Servofrax 




35.9 

35.85 

35.594 

4.5 

0.010 

0.041 

Servofrax 




35.9 

35.85 

35.594 

3.6 

0.020 

0.056 

Servofrax 


2.0-11.0 

6.5 

2.0 

2.0 

1.926 

2 

0.003 

0.006 

Silcon 




4.0 

4.0 

3.876 

2 

0.006 

0.012 

Silcon 




- 

4.0 

3.899 

4 

0.001 

0.005 

Silcon 


6.0-10.0 

8.0 

2.0 

2.0 

1.916 

2 

0.005 

0.009 

Servofrax 




4.0 

4.0 

3.856 

2 

0.010 

0.019 

Servofrax 




4.0 

4.0 

3.886 

4 

0.001 

0.006 

Servofrax 


2.0 

2.0 

1.926 

2 

0.0025 

0.003 

4.0 

4.0 

3.880 

2 

0.005 

0.006 

4.0 

4.0 

3.901 

4 

0.0005 

0.001 


6.0-16.0 11.0 


Germanium 

Germanium 

Germanium 














284 


OPTICAL COMPONENTS 


Table 7-3. Multi-Element Lens Systems Supplied by 
Servo Corporation of America [2]. 


Range 

AX 

(p.) 

Design 

Wavelength 

x<< 

Cm) 

Nominal 

Focal 

Length 

f 

(in.) 

Back 

Focal 

Length 

b.f.l. 

(in.) 

Aperture 

(/7no) 

Circle of 
Confusion 
Xd 
(in.) 

Radius of 
Petzval 
(in.) 

Field of 
View 
(degrees) 

Angular 

Resolution 

(mrad) 








0 

2.00 








1.0 

2.24 

2.0-5.0 

3.5 

2.0 

— 

1.2 

0.004 

5.60 

2.0 

2.63 








4.0 

3.87 

(Fig. 7-la) 







6.0 

5.72 








0 

2.0 








1.0 

2.24 

2.0-5.0 

3.5 

4.0 

— 

1.2 

0.008 

11.20 

2.0 

2.63 








4.0 

3.87 

(Fig. 7-la) 







6.0 

5.72 








0 

0.38 








1.0 

0.43 

2.0-5.0 

3.5 

8.0 

— 

2.0 

0.003 

23.5 

2.0 

0.50 








4.0 

0.75 

(Fig. 7-la) 







6.0 

0.90 








0 

0.55 








1.0 

0.69 

3.5-5.0 

4.5 

4.75 

3.56 

1.2 

0.0026 

23.5 

2.0 

0.90 








4.0 

1.56 

(Fig. 7-lb) 







6.0 

2.52 

6.0-16.0 

11.0 

4.0 

_ 

3 

_ 

_ 

0 

Diffraction 








9.0 

limit 

(Fig. 7-2) 








0.6 

1.2-4.0 

2.5 

3.0 

_ 

2 


_ 

0 

0.40 

(Fig. 7-3) 







6.0 

0.80 

1.2-2.0 

1.6 

3.6 

_ 

1.6 

_ 

_ 

0 

1.1 








6.0 

1.6 

(Fig. 7-4) 







9.0 

2.2 

6.0-16.0 

11.0 

2.2 

_ 

1.5 

_ 

_ 

0 

Diffraction 

(Fig. 7-5) 








limit 



Corning 9752 
Glass 



Glass 


■ iwuv ■//// m \\\uv m 


n 

* 

it, 


; 

r 

! 

i 


j 

h 

H 

nr, 

J 

i 

r 

\w\\w. 

i 

mm///////;///. 

u/iw/nmmwm 


All Lenses 
Germanium 


Fig. 7-4. SCA four- Fig. 7-5. SCA relay lens [2]. 

element achromat [2]. 


Fig. 7-3. SCA three- 
element achromat [2]. 








































































CATADIOPTRIC SYSTEMS 


285 


7.2. Mirrors 

Mirrors in infrared systems, used as reflectors and as image-forming devices, often 
provide the additional function of filtering. Most infrared-system mirrors are front 
surfaced; however, they are back surfaced when used additionally as correctors. Mir¬ 
rors for infrared instrumentation are generally not commercially available, but are 
custom designed for a particular application. Chapter 10 presents the theory and 
equations of reflecting optics. Chapter 8 discusses optical surface coatings, including 
reflective and protective coatings, filter mirrors, and replica mirrors. 

7.3. Catadioptric Systems 

Combined reflecting-refracting, or catadioptric, systems provide correction of aber¬ 
rations over relatively wide angular fields. Table 7-4 and Figs. 7-6 through 7-10 
show various commercially available catadioptric systems. 

Theory and equations of spherical and aspherical concentric optical systems are 
given in detail in Chapter 10. 


Table 7-4. Characteristics of Catadioptric Systems Supplied by Servo Cor¬ 
poration of America (S) and Barnes Engineering Company (B) [2,3]. 


Optical 

System 

Wavelength 

Range 

<M> 

Design 

Wavelength 

(At) 

Nominal 

Focal 

Length 

(in.) 

Aperture 

(/7no) 

Primary 

Diameter 

(in.) 

Lens 

Type 

Field of 
View 
(degrees) 

Angular 

Resolution 

(mrad) 

Supplier 

Maksutov 

0.4-2.0 

1.2 

10.0 

1.6 

_ 

Maksutov 

0 

0.88 

S 

Folded 






quartz 

6 

2.61 


(Fig. 7-6) 






corrector 




Bouwers- 

1.0-4.0 

2.5 

9.6 

1.6 

_ 

Maksutov 

0 

1.5 

S 

Maksutov 






Asj S i 

4 

1.5 


(Fig. 7-7) 






corrector 




Catadioptric 

0.3-0.6 

0.45 

10.1 

3.3 

— 

Maksutov 

0 

0.15 

S 

(Fig. 7-8) 






quartz 

corrector 

6 

0.15 


Folded 

1.0-3.0 


6.0 

1.8 

3.3 

Coming 

2 

1 at 0° 

B 

(Fig. 7-9) 






9752“ 


8 at 10' 


Folded 

1.0-3.0 


4.1 

1.4 

5.3 

Fused 

20 

1 at 0“ 

B 

(Fig. 7-9) 






silica 


8 at 10° 



"Germanate glass. 


Servofrax R 
Corrector 



Fig. 7-6. Maksutov folded optical system 
supplied by SCA [2]. 


Fig. 7-7. Bouwers-Maksutov optical 
system supplied by SCA [2]. 




































286 


OPTICAL COMPONENTS 



Pyrex Mirror 


Fig. 7-8. Catadioptric optical system 
supplied by SCA [2]. 


Secondary Mirror 



Fig. 7-9. Barnes high-speed objective 
system [3]. 



7.4. Interference Filters 

7.4.1. Introduction. Filters are classified by either their transmission character¬ 
istics or the physical phenomena upon which their action is based. A long-wave 
pass filter transmits all radiation with wavelengths greater than the specified value; 
a s ' .'t-wave pass filter passes all radiation with wavelengths shorter than the specified 
value; a bandpass filter transmits only between two wavelengths. 

Some of the physical phenomena that determine filter action are selective reflection 
and refraction, scattering, polarization, interference, and selective absorption. 

7.4.2. Terminology. The description of filters —and even curves of their transmis¬ 
sion—has not been standardized. The terms given below have received some mea¬ 
sure of acceptance, but for precise knowledge of characteristics nothing can substitute 
for a transmittance curve. For the following definitions, refer to Fig. 7-11. 





















































INTERFERENCE FILTERS 


287 



Fig. 7-11. Illustrating filter nomenclature. 


Passband: 


The primary wavelength interval of transmission of a pass filter. 


Stop Band: 


The primary region of reflection, absorption, etc., of a rejection 
filter. 


Background 

Region: The region of low transmission of a pass filter. 


Center The wavelength at the center of the passband. For interference 

Wavelength (\ 0 ): filters, \ 0 is usually the mean of the long- and short-wave cutoffs. 


Halfwidth (HW): The width of the passband at 50% peak transmission — often 

expressed as a percentage of \ 0 . 


Basewidth (BW): The width of the passband of 3% or 1% (or some other small per¬ 
centage) of peak transmittance. It is more precise and useful 
to specify the 3% width, etc., as (AA.V 03 ; thus HW becomes (AA) 0 .5. 


Peak Trans- The maximum transmittance in the passband. For interference 

mittance (T 0 ): filters this is often specified as a percent of the uncoated substrate. 


Substrate Trans¬ 
mittance (T g ): Substrate transmittance. 

Free Filter Range The wavelength interval over which the background is less than 
(FFR): a specified amount except where the passband is. 


Cutoff and Cuton 

Wavelengths (A c ): The wavelengths of the limits of the passband. 

Slope: The linear approximation to the cuton or cutoff slope, expressed 

as the ratio (\ 08 — A c )/A c , where Ao.s is the wavelength of 80% 
transmittance. 


The user of filters should take care to understand the nomenclature used by the 
individual manufacturers. In particular he should note whether To is given in terms 
of the ratio of the filter transmission to the substrate transmission, whether \ c is 
given as a 5%, 3%, or 1% cutoff, and whether FFR is specified for 1%, 0.1%, etc. 













288 


OPTICAL COMPONENTS 


7.4.3. General Theory of Interference Filters. 


Phase Difference (8): The phase difference in radians is 2rrant. 


Wavenumber (cr): The reciprocal of wavelength. 


Dimensionless Some design (set) wavelength divided by the variable wave- 

Wavenumber (g): length, A. 0 /A.. 


Quarter-Wave Optical The thickness of a layer measured in the number of quarter 
Thickness (QWOT): waves of a design wavelength. Thus, QWOT = Ant. 


H and L Layers : Quarter-wave thicknesses of the higher index in a stack of 

layers are often written H; quarter-wave thickness of lower 
index as L. 


Quarter-Wave Stack : A periodic array of alternating H and L layers, viz., HLHLHL 

= (HL) 3 or LHLHLH = ( LH ) 3 . 


High-Reflectance Zone: Region of high reflectance as shown in Fig. 7-12. 

Quarter-wave stacks are the most simple from the design standpoint. On a fre¬ 
quency scale they have the symmetric transmission structure shown in Fig. 7-12. 
The width of the high reflectance zone is given by 


kg 


4 

= — arc sin 

7 T 


n/i — n L 
ni, + n L , 


The maximum reflectivity is given by 


P + P- 1 - 2 
~~ P + P~ l + 2 


For P > > P 1 , 


P = 


ni n t - 2 

_m-x ni-s 


n£\ 2 ru 
n J n s 


P = 


ni 

.ni-i 


2 1 



n 0 n s 


R 


max 


P-4 

P 


l even 

l odd 


The theoretical curve given in Fig. 7-12 is useful for beginning calculations. Figure 
7-13 shows the transmission of a quarter-wave stack over a larger range of g. 

Stacks with unequal optical thickness ratios can also be very useful. The 2:1 stack 
has the configuration 


LLH LLH LLH • • • = {LLH) m 


The first-order high-reflectance zone occurs at g = 1, when LLH is X 0 /2. The second- 
order zone occurs when g = 2 and LLH is k 0 . There is no high reflectance when g = 3 
because LLH is 3 A«/2; thus H is a half wave and LL is a full wave, and all LLH’s are 
absentee layers. 

The general p:q stack can be analyzed in a similar way. The following features 
may be useful: 

1. The high-reflectance zone may not be a center of symmetry 

2. The high-reflectance zone of a quarter-wave stack is wider than other p:q stacks. 












INTERFERENCE FILTERS 


289 


NOISSIWSNVHX 



AUAIJ.D3333H 


Fig. 7-13. Computed spectral reflectivity of an eight-layer quarter-wave stack (-) and its envelope 

of maximum reflectivity (-). 












290 

3. 


OPTICAL COMPONENTS 


The number of oscillations outside the high-reflectance zone increases as the 
number of layers is increased. 

4. Other things being equal, the quarter-wave stack has the highest reflectivity. 

Additional layers can be added to: (1) increase the reflection in the stop band, or 
(2) decrease the reflection in the passband. To increase reflectivity use H layers at 
both ends of the basic period: 

HLHLHLH 

To decrease the reflection, the layer can be replaced by its Herpin equivalent and anti¬ 
reflection coatings designed for the desired wavelength. The stack can also be varied 
by computer techniques based on variational principles. 

7.4.4. Long-Wave Pass Interference Filters. The design of a long-wave pass 
interference filter is based on Fig. 7-12. Low-reflectance regions are regions of high 
transmission, and can be designed on that basis. The curves change with different 
ratios of n H to n L and different numbers of layers. The design proceeds by choosing 
a useful substrate and a design X 0 . Some changes can then be made. The long-wave 
cutoff is determined by either the substrate absorption or the second-order maximum 
(Fig. 7-13). Commercially available long-wave pass filters are described in Sec. 7.9. 
They have the following properties: 

1 . The slope of the cuton increases with the number of layers. 

2. The maximum reflectance increases with the number of layers. 

3. The width of the reflectance zone increases as n H /n.L increases. 

4. A higher-order reflectance peak has a sharper cuton but a narrower transmission 
region (AX). 

5. Angle shift is minimized by high values for and n L or by more high-index 
materials in the basic period. 

7.4.5. Short-Wave Pass Interference Filters. The comments applicable to long¬ 
wave pass filters (Sec. 7.4.4) also apply here. Short-wave pass filters are usually de¬ 
signed from quarter-wave stacks because these have the longest region of high trans¬ 
mission to the short-wave (high-frequency) side of the high reflectance zone. Then 
an antireflection coat is applied to the stack in the transmission region. Some com¬ 
mercially available filters are described in Sec. 7.9. 

7.4.6. Bandpass Interference Filters. Every filter is a bandpass filter. If the 
desired pass region is smaller than that obtained by a long-wave pass filter, a short¬ 
wave pass can be added; they can be deposited on opposite sides of the substrate. Nar¬ 
rower bandpasses are obtained by interference techniques, similar to that for the 
Fabry-Perot interferometer. The transmission is given by 


(l - \'R,R- t f 

where T i, T 2 , R i, and Ri are the transmittance and reflectance of the plate coatings 
(looking from the gap); €i and e 2 are the phase shift upon reflection, and nt is the optical 
path of the gap. The transmission is a sinusoidal function of 1/X or a. The region 
between adjacent transmission peaks is the free spectral range. In filter language 
this is the free filter range, or FFR. This spacing ay is given by 


1 + 


4V^sin(^-ii±- e! 


(i - VR^ 2 y 


i 











INTERFERENCE FILTERS 


291 


The narrowness of a line is given by the Q: 

ko cr o 

= ——-— =-=- ; - rmr 

(A\)o.5 (Ao-)o.s i _ VR.Ri 

If R i and R 2 are large and if e x and e 2 are constant over A A., then the line has a Lorentz 
shape. Narrowband filters can have the following construction 



HLH LL HLH 

This can be thought of as two filters separated by a half-wave of low index material. 
The curve for this filter is shown in Fig. 7-14. These filters are usually combined with 
blocking filters to isolate the narrow band. Some commercially available filters are 



0.78 0.79 0.80 0.81 0.82 

WAVELENGTH (jn) 

Fig. 7-14. Measured transmission of a narrowband filter. 


7.4.7. “Square-Band” Interference Filters. This type of filter-not really a 
square band-can be designed as a general p.q stack. The design is 

(HL) m LL (HL) m LL • • • LL (HL) m 

The filter is generally steeper and has a rippled top. The rejection is also better than 
the quarter-wave stack. A comparison of the "square” filter with the normal quarter- 
wave stack is given in Fig. 7-15. These can also be thought of as multiple-wave filters. 

7.4.8. The Filter Matrix. The relations between the electric and magnetic fields 
on two sides of the ith interface are given by 


E 

H 


= Mi 


E' 

H' 
















292 


OPTICAL COMPONENTS 



Fig. 7-15. Measured transmittance of two bandpass filters with 
nominal 2% halfwidth at 4.29 fi. 


Here M , is the characteristic matrix of the ith surface 




cos 8i jn ~i 


sin 




Mi is given by 

sr 

S, 


The relation between 


is 


E o 
Ho 


and 


E m 

H m 


Eo 

m 

E m ~ 


Em 


a ii 

ja 12 

Em 


= n 

Mi 

= M 


= 




H 0 _ 

i = l 

H m 


H m _ 


Ja 2 1 

a 22 

H m _ 


The determinant \M\ is 1, so that the reflectivity R can be calculated from a knowledge 
of three elements. 

For any periodic layer, the period can be reduced to a fictitious bilayer, and if the 
period occurs m times, then 

Eo 1 

= (Af iM 2 ) m 
Ho J n s 

For a symmetrical layer the period can be replaced by a monolayer. 




















INTERFERENCE FILTERS 293 

7.4.9. The Herpin Equivalent Layer. The thickness of the layer d n can be written 
in terms of the phase 

= 360 ntl\ (degrees) 

= 2irntl\ (radians) 

= nt/\ (wavelengths) 

= arc cos a 11 
n h = a 2 i/(l - a u 2 ) 112 

7.4.10. Analogies with Transmission-Line Theory [41. The matching theorems 
involving calculation of line admittance, characteristic admittance, reflection coeffi¬ 
cient, etc., can be applied to optical multilayers by treating the refractive index as 
the admittance. Some useful equations are: 

For the nth element of an infinite lumped-constant line (s = series; sh = shunt): 

i n = Ae yn 

Z, 

coshy = 1 + —— 


e y = 1 + 




Z„ = ± + (Z s /2) 2 

sinhy = Z 0 /Z stl 

For an infinite distributed-parameter line: 

Z = R + j(oL Y — G -f j(oC 


z 0 = ± V z/Y y = ± Vzy = z 0 Y 

For a line terminated by Z«: 

D — Z ° ^ K — Y H ~ 

Zo + Z R Y r + Yo 


Z(l) = Zo 


Z 0 sinh yl + Zr cosh yl 
Zo cos yl + Z R sinh yl 


ZciZ 0 p — z o 2 

where Z c i is the short-circuit impedance and Z„ p is the open-circuit impedance. 

Impedance matching: A section of lossless lumped line can be chosen to join generator 
and load for maximum power transfer if 

Zo 2 = ZinZterm 


and the length is a quarter wave. 

7.4.11. Effects of Angle of Incidence. The transmission band is a function of the 
angle of incidence. An effective optical thickness can be used. 

( nt)eff— nt cos 6 

where 6 = angle of incidence. This technique can be used for a few layers, but since 
the angle of incidence for each layer is a function of the original angle and all the 












294 


OPTICAL COMPONENTS 


preceding layers, the technique is cumbersome. Substituted into the matrix formu¬ 
lation, however, the effective optical thickness is again useful. For nonnormal inci¬ 
dence the filter also becomes polarizing; for simple layers the standard equations for 
reflectance are useful, or they can be put in matrix form for iteration. Angle effects 
can be minimized by the use of higher index or using more material of a higher index 
in a layer. The effective thickness is shown in Fig. 7-16. 

Film Index 

4 3 2.5 2 1.5 1.25 



EFFECTIVE FILM THICKNESS/FILM THICKNESS 


Fig. 7-16. Effective film thickness [5]. 


7.4.12. Effects of Temperature. Cooling a filter changes both the actual thickness 
of layers and the refractive index. Thus the filter will change its center wavelength 
as the optical path changes: 


Ant At 
~AT = n AT 


+ t 


An 
A T 


These terms are calculable from data of thermal expansion and refractive index change 
with temperature given in Chapter 8. 

7.4.13. Substrates and Films. Table 7-5 is a list of the commonly used substrate 
materials. Chapter 8 gives the physical data for these materials. Sometimes it will 
be necessary to extrapolate the data to thinner samples. 


Table 7-5. Commonly Used Substrate Materials 


Material 

Refractive Index 

Transmission Range 

(a*.) 

Irtran 1 

1.38-1.31 

1-8.5 

Lithium fluoride 

1.39-1.30 

0.2-9 

Calcium fluoride 
(also as Irtran 3) 

1.44-1.32 

0.2-12 

Vycor 

1.46 

0.25-3.5 

Fused quartz 

1.48-1.41 

0.2-4.5 

Barium fluoride 

1.51-1.41 

0.2-15 

Glass 

1.70-1.51 

0.32-2.5 

Magnesium oxide 

1.77-1.62 

0.3-8.5 

Sapphire 

1.83-1.59 

0.2-6.5 

Irtran 2 

2.29-2.15 

1-14.5 

Irtran 4 

~2.4 

2.0-24 

Arsenic trisulfide glass 

2.66-2.37 

0.6-11 

Silicon 

3.50-3.42 

1.2-15 

Germanium 

4.10-4.00 

1.8-23 







CHRISTIANSEN FILTERS 295 

Table 7-6. Commonly Used Film Materials. [6] 

Material Refractive Index Range of Transparency 9 Comments 




From 

To 


Cryolite 

1.35 

<200 mg. 

10 g 

1 

Chiolite 

1.35 

<200 mg 

10 g 

1 

Magnesium fluoride 

1.38 

230 mg 

5 g 

2,3 

Thorium fluoride 

1.45 

<200 mg 

10 g 

— 

Cerium fluoride 

1.62 

300 mg 

> 5 g 

4 

Silicon monoxide 

1.45 to 1.90 

350 mg 

8 g 

5 

Sodium chloride 

1.54 

180 mg 

>15 g 

6 

Zirconium dioxide 

2.10 

300 mg 

> 7 g 

2 

Zinc sulfide 

2.30 

400 mg 

14 g 

7 

Titanium dioxide 

2.40 to 2.90 

400 mg 

> 7 g 

8 

Cerium dioxide 

2.30 

400 mg 

5 g 

2,3 

Silicon 

3.50 

900 mg 

=4 

oo 

— 

Germanium 

3.80 to 4.20 

1400 mg 

>20 g 

— 

Lead telluride 

5.10 

3900 mg 

>20 g 

— 

Notes: 





1. Both materials are sodium-aluminum fluroide compounds, but differ in the ratio of Na to A1 and have different 
crystal structure. Chiolite is preferable in the infrared, because it has less stress than cryolite. 

2. These materials are hard and durable, especially when evaporated onto a hot substrate. 

3. The long wavelength is limited by the fact that, when the optical thickness of the film is a quarter-wave at 
5 /x, the film cracks because of the mechanical stress. 

4. Other fluorides and oxides of rare earths have refractive indices in this range from 1.60 to 2.0. 

5. The refractive index of SiOx (called silicon monoxide) can vary from 1.45 to 1.90 depending upon the partial 
pressure of oxygen during the evaporation. Films with a refractive index of 1.75 and higher absorb at wave- 


lengths below 500 m /x. 

6. Sodium chloride is used in interference filters out to a wavelength of 20 /a. It has very little stress. 

7. The refractive index of zinc sulfide is dispersive. 

8. The refractive index of Ti0 2 rises sharply in the blue spectral region. 

9. The range of transparency is for a film of quarter-wave optical thickness at this wavelength. These values 
are approximate and also depend quite markedly upon the conditions in the vacuum during the evaporation 
of the film. 


Table 7-6 is a list of commonly used film materials [6]. The data for thin evaporated 
films are often different from those of Chapter 8, which are for solid samples. Since 
the values vary with deposition conditions, only representative numbers are given in 
Table 7-6. 

7.5. Christiansen Filters [7,8,9] 

These filters are made of small, closely packed particles of an infrared-transparent 
substance suspended in a liquid or a gas. The optical properties of the materials are 
so chosen that the indices of refraction of the particles and the suspending medium 
are the same at the wavelength that is to be transmitted. The dn/d\ values of the 
liquid and the solid particles are chosen to be as widely different as possible. Thus, 
as the wavelength is progressively increased or decreased from the wavelength at which 
equality of the indices occurs, the difference in index between the particles and the 
suspending medium increases rapidly. 


OPTICAL COMPONENTS 


296 

One form of Christiansen filter for the infrared is obtained by using quartz particles 
in air. Figure 7-17 shows the dispersion curve of ordinary quartz. The refractive 
index is unity at 7.4 g. At this wavelength, therefore, the quartz particles have the 
same refractive index as air, and high transmission occurs. 

Table 7-7 lists other materials which when suspended in air, can be used as Christian¬ 
sen filters. Also shown are the wavelengths at which maximum transmission occurs. 
These are designated as Christiansen wavelengths. 



Fig. 7-17. Dispersion curve of quartz, 
showing Christiansen wavelength [7]. 


Table 7-7. Christiansen Wavelengths 
of Selected Materials [71 


Crystal in Air 

Christiansen 

Wavelength 

(g.) 

Quartz 

7.3 

LiF 

11.2 

MgO 

12.2 

NaCl 

32 

NaBr 

37 

Nal 

49 

KBr 

52 

KI 

64 

Rbl 

73 

Til 

90 


Figure 7-18 shows the effect of quartz powder in a medium of pure CCh and CS 2 
as compared with quartz in air. Because values of dnldT are relatively high, these 
filters are sensitive to temperature fluctuations. By the same token, in a controlled 
environment the center of the passband can be varied by changing the temperature. 






SELECTIVE REFLECTION FILTERS 


297 



Fig. 7-18. Position of the Christiansen peak for 
quartz power in liquids, (a) Quartz in a 50% by vol¬ 
ume mixture of CS 2 and CC1 4 . (6) Quartz in pure 
CC1 4 . (c) Quartz in air [10]. 


7.6. Selective Reflection Filters [11] 

Selective reflection filters are made of crystalline materials that show selective 
reflection at certain wavelengths. These filters are useful to about 200 /u,. 

In practice, radiation from a suitable source is collimated and directed at the sur¬ 
face of a polished crystal whose residual ray occurs at the wavelength to be selectively 
reflected. After three or four successive reflections from similar crystal plates, only 
the residual ray is present with any appreciable intensity, the other wavelengths 
having been attenuated by a factor of several thousand. Figures 7-19 and 7-20 indicate 
the wavelengths of the residual rays of a number of materials. 



Fig. 7-19. Reststrahlen (residual ray) frequencies of alkali halide crystals. 
























298 


OPTICAL COMPONENTS 



7.7. Selective Refraction Filters 

Filtering by selective refraction depends upon the dn/dk of a lens material. Radi¬ 
ation of different wavelengths will be focused at different points along the optical axis 
if the lens is used in a wavelength region where it possesses high dispersion. This 
technique is particularly useful when the lens is used near an absorption band, because 
the refractive index will be considerably different on opposite sides of the band. This 
method of focal isolation or selective refraction is illustrated in Fig. 7-21. 



Fig. 7-21. Filtering by the method of focal isolation. 


Two quartz lenses are usually employed in this method. The refractive index of 
quartz in the near infrared is about 1.5, and about 2.15 in the range 60 /a to 100 /a. 
(Quartz absorbs in the region around 9/a.) In Fig. 7-21, radiation in the far infrared 
is passed through the aperture, brought to a focus at aperture A', and transmitted. 
Visible and near-infrared radiation, being deviated less, impinges upon screen No. 2. 
Lens No. 2 focuses the desired radiation upon the detector. The two opaque discs, 
d\ and d z , obscure the paraxial zone of the lenses and prevent transmission of direct 
radiation. 

7.8. Polarization Interference Filters [11] 

The polarization interference filter, sometimes called a Lyot-Ohman (or birefringent) 
filter, isolates a spectral band only a few angstroms wide. These filters are constructed 
of alternate plates of polarizers and birefringent crystals (e.g., quartz) as shown in 
Fig. 7-22. 















COMMERCIALLY AVAILABLE FILTERS 


299 


Crystal Quartz Plates 



Fig. 7-22. Birefringent filter [11]. 

The birefringent crystal and quartz plates are cut with their optical axes parallel 
to the large faces. The axes of the polarizers are oriented at 45° to the quartz optical 
axes. Linearly polarized light, incident on the first quartz plate at 45°, would have 
its plane of polarization rotated by 90° if the plate were a half-wave plate (or if the optical 
path difference between the ordinary and extraordinary rays were any odd multiple 
of half-waves). If the plane of polarization is rotated at 90°, the radiation will not 
be transmitted by the second polarizer. Since the phase difference introduced between 
the ordinary and extraordinary rays depends on wavelength as well as on the thickness 
of the quartz plate, the same plate may be a 5/2-wave plate for one wavelength, a 7/2- 
wave plate for another wavelength, and 9/2-wave plate for still another wavelength. 
Each of these wavelengths will be blocked by the polarizer following the plates, whereas 
those wavelengths for which the difference in optical path between the two polariza¬ 
tions is an even number of half waves will be completely transmitted. 

Figure 7-23(a) illustrates the transmission of the first quartz plate and its polarizers. 
If each quartz plate is made twice as thick as the preceding one, it will have twice as 
many transmission maxima and minima in a given wavelength interval. The trans¬ 
mission curves for the second, third, and fourth plates are illustrated in Fig. 7-23 (6), 
(c), and id). The transmission of the entire filter is the result of all these transmission 
curves and is shown in Fig. 7-23(e). 



Fig. 7-23. Transmission of Lyot filter 
and its components [11]. 


7.9. Commercially Available Filters 

Figure 7-24 presents transmittance curves of typical Bausch & Lomb filters. Fig¬ 
ure 7-25 shows transmittance curves of interference filters available from Eastman 
Kodak. Figure 7-26 shows transmittance curves of interference filters available 
from Infrared Industries, Inc. Figure 7-27 shows transmittance curves of filters 
available from Optical Coating Laboratory, Inc. Other infrared filter manufacturers 
include Optics Technology, Inc., Fish Schurman for Schott Glasswerke, Baird-Atomic 
(mainly for visible and ultraviolet), Farrand Optical Company, Inc.; a number of other 
companies are listed in the Optical Industry catalog. 






























300 


OPTICAL COMPONENTS 



Bandpass filter (cover glass 
substrate) Xg = 2.45 p 



Long-wave pass filter 
(cover glass substrate) 
*0 = 1.8 p 



Bandpass filter in series with 
auxiliary long-wave pass filter 
(cover glass substrate) Xq = 2.48 p 



Bandpass filter (cover glass 
substrate) Xq = 2.26 p 



Bandpass filter (cover glass 
substrate) Xq = 4.42 p 



Bandpass filter (cover glass 
substrate) Xq = 1.68 p 



Bandpass filter (cover glass 
substrate) Xq = 3.82 p 



Bandpass filter (cover glass 
substrate) Xq = 1.89 p 



Bandpass filter (cover glass 
substrate) Xq = 3.95 p 



Bandpass filter (cover glass 
substrate) Xq = 1.94 p 


Fig. 7-24. Transmittance of Bausch & Lomb infrared 
interference filters [4], 














































COMMERCIALLY AVAILABLE FILTERS 


301 



Bandpass filter (cover glass 
substrate) X Q = 3.74 p 



Bandpass filter (cover glass 
substrate) Ag = 0.175 p 



Bandpass filter (cover glass 
substrate) Ag = 3.01 p 



Long-wave pass filter 
(calcium fluoride substrate) 
Ag = 3.4 p 



Bandpass filter (Ag = 3.45 p) 
in series with long-wave pass 
filter (Aq = 2.6 p) 



Long-wave pass filter 
(magnesium oxide substrate) 
Ag = 3.44 p 



Long-wave pass filter 
(cover glass substrate) 
Ag = 1.9 p 


A-2 Filters in Series 
B-Primary Filter 



Long-wave pass filter (Ag = 2.44 p) 
in series with long-wave pass 
filter (Ag = 1.5 p); cover glass 
substrate 


Fig. 7- 24 ( Continued ). Transmittance of Bausch & Lomb 
infrared interference filters [4]. 









































EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE 


302 


OPTICAL COMPONENTS 


6 s ? 


100 r (a) 


rx 


50 


Dotted Line indi¬ 
cates E. K. type- 
301 coating opens 
up again beyond 

; \: 


J L 


0.4 0.6 0.8 1.0 

WAVELENGTH (p) 


1.2 



Short-wave pass filter 
(plate glass substrate with 
Kodak No. 301 coating) 


Short-wave pass filter (heat¬ 
absorbing glass substrate 
with Kodak No. 301 coating) 



Long-wave pass filter; Aq = 1.0 p 



Long-wave pass filter; Aq = 2.0 p 




WAVELENGTH (p) WAVELENGTH (p) 

Long-wave pass filter; \q = 3.0 p Bandpass filter; Aq = 2.2 p 

Fig. 7-25. Transmittance of Eastman Kodak infrared 
interference filters [1], 




















EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE (%) 


COMMERCIALLY AVAILABLE FILTERS 


303 




Bandpass filter; Ag = 2.7 p Bandpass filter; Ag = 4.5 p 




Bandpass filter; Ag = 5.75 p Bandpass filter; Ag = 9.7 p 




Bandpass filter; Ag = 10.9 p Bandpass filter; Ag = 11.0 p 

Fig. 7-25 ( Continued ). Transmittance of Eastman Kodak 
infrared interference filters [1]. 


















304 


OPTICAL COMPONENTS 



Far-infrared, long-wave pass filters (silver 
chloride substrate) 



Far-infrared, long-wave pass filters (silver 
chloride substrate plus polystyrene protective 
coating) 


Fig. 7-25 ( Continued ). Transmittance of Eastman 
Kodak infrared interference filters [1], 










EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE (%) EXTERNAL TRANSMITTANCE (%) 


COMMERCIALLY AVAILABLE FILTERS 


305 



(a) 



(b) 



(c) 

Fig. 7-26. Transmittance of Infrared Industries inter¬ 
ference (Infratron) filters, (a) Long-wave pass; (6) spike; 
(c) long-wave pass combined with detector [12]. 




























306 


OPTICAL COMPONENTS 





100 


z 

o 

HH 

CO 

CO 

HH 

s 

w 

s 

H 

es 


801— 

60 

40 

20 

0 


Half 

Band - 
Width 

Attenuation 

Area 




Attenuation 

Area 


2.0 


(c) 


3.0 

microns 

(d) 


4.0 


Fig. 7-27. Transmittance of Optical Coating Labs infrared interference filters. 


7.10. Absorption Filters [7,12,13] 

Many materials in solid, liquid, or gaseous state, including those discussed in Chap¬ 
ter 8, can be used as selective absorption filters in various regions of the infrared 
spectrum. These filters have high transmission above or below a certain wavelength 
where high absorption produces a sharp cutoff or cuton. 

Long-wave pass filters in the near infrared are normally made of plastic materials 
containing dyes, colored glass, or sublimated phthalocyanines upon glass. Other 
long-wave pass filters consist of glass coated with plastic dye solutions. Figure 7-28 
shows the characteristics of these filters [14]. 

Figure 7-29 shows the variety of cuton wavelengths that are obtained with various 
semiconductors. By proper doping the location of the absorption limit can be moved, 
although this reduces the gradient. A similar effect can also be obtained by the use 
of mixed crystals, Fig. 7-33, although no such filters are available commercially at 
present. An excellent reference on mixed-crystal semiconductors is R. H. Bube [16]. 













PRISMS 


307 



Fig. 7-28. Near infrared dyed-plastic filter characteristics [14], 



Fig. 7-29. Transmission of selected semiconductors [15]. 


7.11. Prisms 

Prisms are used principally for deviating light or dispersing it. Any prism does 
both, but it can be arranged to maximize one but minimize the other. Figure 7-30 
illustrates the geometry and defines the symbols. 



The angular magnification Me is 

— cos 0i cos 0 2 ' 
A1 q = “ ~z 

cos do cos 6\ 

The total deviation is 

8 = 0, + 02 - a 

























308 


OPTICAL COMPONENTS 


The deviation can also be expressed by 

sin 0i = sin a k/(n 2 2 /ni 2 ) — sin 2 0 2 — cos a sin 0 2 
Minimum deviation is given by 

Sinj (8 min a) 

n 2 = n,-:-—- 

sin a/2 

The change in deviation with wavelength, or the dispersion is 

d8 dn 2 2 sin a/2 

^ ^k x/l — (n 2 ln i) 2 sin 2 a/2 


The resolving power is 


k dd dn 

— = a— -— 

dk dn dk 


dn 



7.11.1. Dispersing Prisms. Figure 7-31 illustrates examples of various types of 
dispersing prisms, commonly used in spectroscopy. These are the constant deviation 
type. The Wadsworth prism, operating at minimum deviation, is used extensively 
in infrared monochromators, where for a constant angle between the collimator and the 
telescope, astigmatism-free images can be obtained at various wavelengths by rotating 
the prism. The Littrow prism reflects the light directly back along the direction from 
which it came [18,19]. 


45 ° 



(a) Constant-Deviation Prism 



(b) Abbe Prism 



Mirror 


(c) Wadsworth Prism (d) Littrow Prism 

Fig. 7-31. Dispersing prisms. 

7.11.2. Deviating Prisms [18]. Figure 7-32 shows various types of total reflecting 
deviating prisms. As shown in Fig. 7-32(a), rays enter perpendicular to one of the 
shorter faces of a total reflection prism, are totally reflected from the hypotenuse, and 
leave at right angles to the other short face. Such a prism can be used in two other 
ways, as shown in ( b ) and (c). The Dove prism, (c), interchanges the two rays, and if 
the prism is rotated about the direction of the radiation the rays rotate around each 
other with twice the angular velocity of the prism. The roof prism, (d), is similar 



















DIFFRACTION GRATINGS 


309 



(a)^Total Reflection (b) Porro 


(c) Dove or Inverting 




(e) Triple Mirror 


Fig. 7-32. Deviating prisms [18]. 


to the total reflection prism (a), except that it introduces an extra inversion. The 
triple mirror, (e), is made by cutting off the corner of a cube by a plane which makes 
equal angles with the three faces intersecting at that corner. It has the useful prop¬ 
erty that any ray striking it will, after being internally reflected at each of the three 
faces, be sent back parallel to its original direction. 

7.11.3. Prism Materials. The most common prism materials for the infrared region 
include crystal quartz, rock salt, potassium bromide, lithium fluoride, and calcium 
fluoride. These materials are especially popular because of their availability as 
synthetic crystals (except for quartz, in general) [20], 

Materials suitable for operation in the wavlength region beyond 15 /x, include silver 
chloride, thallium-bromo-iodide (KRS-5), cesium bromide, and cesium iodide. 

The choice of an infrared prism material depends upon such characteristics as trans¬ 
mission, refractive index, and dispersion as a function of wavelength and possibly of 
temperature. In addition, the mechanical, physical, and chemical properties must 
also be considered. The properties of the aforementioned materials are discussed in 
Chapter 8. 

7.12. Diffraction Gratings 

A diffraction grating of the transmission type consists of a large number of small, 
equal-size, equally separated slits; each slit causes a diffraction pattern, and the waves 
from the individual slits also interfere to form a combined interference-diffraction 
pattern. The intensity can be written as 

/„ (area) 2 sin 2 (3 sin 2 Ny 
I \ 2 D 2 (3 sin 2 y 

77-a sin 6 ., . „ , , . 

where (3 = ---; a = slit width, 6 = angle to image point 

A 

Trd sin 6 . 

y — ---; d = slit spacing 

A 

/o = intensity at grating 

D = grating-image distance 

































310 


OPTICAL COMPONENTS 



Fig. 7-33. Transmission of some mixed crystals. 


The grating equation is 


mk = d(sin 0 — sin O') 

where m = order of interference 


8 = angle of incidence 
0' = angle of diffraction 
d = slit separation 
The angular dispersion is 

d9'Id = (mid cos O') 

The resolving power is 

— = (k/dk) = mN 
kv 

where v = wavenumber. 


7.12.1. Blazed Gratings. It is possible to "blaze” a grating by ruling its grooves 
so that its sides reflect a large fraction of the incoming radiation of suitably short 
wavelengths in one general direction. Controlled groove shape is especially important 
in the gratings known as echelettes and echelles. In these gratings the grooves are 
ruled with one face optically flat. This face is inclined at an angle </> (see Fig. 7-34) 
to reflect or refract most of the incident radiation in a desired direction. In this way, 
the grating concentrates radiation in a particular spectral order, producing a brighter 
image than an ordinary diffraction grating. 



(a) Reflecting Grating (b) Transmission Grating 

Fig. 7-34. Reflection and transmission of gratings [18,19], 
















DIFFRACTION GRATINGS 311 

7.12.2. Concave Gratings. This type of grating consists of a concave mirror 
with ruled lines spaced equally along a chord across its center (Fig. 7-35). Radiation 
that passes through a slit and falls on such a grating is dispersed by it into spectra, 
in accordance with the standard grating formula. The concave grating requires no 
separate collimator or objective, since it is both a dispersing and a focusing element. 



Fig. 7-35. Rowland circle focal curve 
of a concave grating [21]. 


Concave gratings have the advantage of focusing the radiation, thus eliminating 
the necessity for auxiliary lenses which introduce aberrations although the gratings 
suffer aberrations, principally astigmatism. 

The focus for the spectrum from a concave grating is given by the formula 

cos i cos 2 i \ 

~P S~) + 

where i and r are the incident and reflected angles 

p is the radius of curvature of the concave blank 
S is the slit-to-grating distance 

S ' is the distance from the grating to the image of the spectrum 

This equation can be satisfied by setting S = p cos i and S ' = p cos r, the polar equa¬ 
tions for a circle of diameter p, containing the points S, S and G, which is shown in 
Fig. 7-35 as the familiar Rowland circle. 

7.12.3. Ebert-Fastie Plane Grating Mountings [21]. Figure 7-36(a) shows the 
original Ebert [22] mounting, which was modified by Fastie [23] as shown in Fig. 
7-36(6). Both mountings can be designated as "side-by-side” designs; the entrant 
rays are on one side of the grating, the emergent rays on the other. Figure 7-36(c) 
is an "under-over” design. The entrant rays pass below the grating and the emergent 
rays pass above. 

7.12.4. Concave Grating Mountings [21]. Figure 7-37 shows different mountings 
for the concave grating designed to maintain slit, grating, and plate- or film-holder 
on the Rowland circle (Fig. 7-36). 

The Rowland mounting, Fig. 7-37(a), utilizes the geometric principle that the locus 
of the apex of a series of right triangles having a common hypotenuse is a circle having 
the hypotenuse as a diameter. The grating, G, and plateholder, P, are fixed on opposite 
ends of a rigid bar, which forms the hypotenuse and the diameter of the Rowland 
circle. The slit, S, is placed at the intersection of two tracks, SP and SG, which are 
carefully constructed at right angles. Rollers under G and P are constrained to travel 
along SG and SP, so that as the bar, GP, is moved, the grating remains on the optical 


cos r cos 2 r\ 

~p SW = ° 









312 


OPTICAL COMPONENTS 



(0 


Fig. 7-36. Plane grating mountings; (a) original Ebert design; ( b) Fastie 
modified "side-by-side” design; (c) Fastie modified "over-under” design [21]. 


(a) Rowland 



(c) Abney 


(e) Beutler Radius 


(g) Seya-Namioka 



(d) Eagle 


(f) Wadsworth 


(h) Grazing Incidence 


Fig. 7-37. Concave grating mountings [21]. 


axis fixed by the condensing system, and the Rowland circle moves to coincide with 
all three optical elements [24]. 

The Paschen-Runge or Paschen mounting, Fig. 7-37(6), usually consists of large 
circular tracks 21 or 35 feet in diameter, which are built in a room that can be temper¬ 
ature stabilized and darkened. Plateholders can be clamped to these tracks where 
desired. The slit mounts protrude through a wall into an adjoining room where the 
excitation equipment and light sources are located [25]. 




































DIFFRACTION GRATINGS 


313 


In the Abney mounting, Fig. 7-37(c), the plateholder is mounted on the normal 
to the grating. The slit is mounted on an arm rotating about the center of the Row¬ 
land circle to change the wavelength range. The slit assembly must also be rotated 
about an axis below the slit opening itself, so that the source-slit axis will still point 
at the grating [26]. 

For the Eagle mounting, Fig. 7-37(c0, three independent mechanical adjustments 
are required to maintain the elements on the Rowland circle. 

(1) The grating must be rotated to change the wavelength range. 

(2) The grating must be moved along the optical axis to maintain the focus. 

(3) The plateholder must be tilted about an axis under the slit to remain in focus 
across its length [27]. 

In the Beutler radius mounting, Fig. 7-37(e), the slit and plateholder are mounted 
permanently on the Rowland circle, and the grating is mounted on an arm which rotates 
about the center of the Rowland circle [28]. 

For the Wadsworth mounting, Fig. 7-3 7(f), the focal distance is half the radius 
of curvature, the individual elements are not located on the Rowland circle, and a second 
optical element is introduced in the form of a concave collimating mirror. This arrange¬ 
ment produces a stigmatic image and a linear dispersion, as in the Rowland and Abney 
mountings. The concave mirror is mounted at its focal distance from the slit, tilted 
slightly off axis to irradiate the grating with parallel light. The grating may be located 
either side of the optical axis, as close to it as possible to minimize aberrations. The 
plateholder is located on a bar whose axis forms the normal to the grating. To change 
the range to higher wavelengths, the bar is rotated away from the mirror about an 
axis under the grating face. The plateholder must be moved away from the grating 
to remain in focus. In any position of the plateholder the shorter wavelength radiation 
is nearest to the concave mirror [29]. 

Seya-Namioka mounting, Fig. 7-37(g), is based on the principle that if the angle 
between the entrant and emergent rays in a spectrometer is 70° 15' there will be such 
slight defocusing, if the grating is merely rotated about its own vertical axis, that the 
image would be entirely acceptable for scanning monochromator usage [30]. 

Grazing incidence mounting, Fig. 7-37 {h), [31], operates on the principle that, if 
a grating is illuminated at grazing incidence, the short-wavelength (below 1000 A) 
radiation will be totally reflected. Angles of incidence as high as 85-89° have been 
employed to observe wavelengths as low as 53 A [32] or even 12.1 A [33]. 

7.12.5. Production of Gratings. Gratings are engraved by highly precise ruling 
engines which use a diamond tool to press a series of many thousands of fine shallow 
burnished grooves into a smooth metallic surface. 

Gratings for the range 1500 to 10,000 A are commonly ruled with 5000 to 30,000 
grooves per inch (the usual value is near 15,000), on a thin layer of aluminum de¬ 
posited on glass by evaporation in vacuum. Gratings for the infrared region are also 
ruled on gold, silver, copper, lead, or tin mirrors with coarser groove spacings. 

Gratings of 2-in., 4-in., or 6-in. ruled width are commonly used in commercial spec¬ 
trographs, with projection distances of 20-180 in. In large research instruments, 
gratings of 6- to 10-in. ruled width are used with projection distances of 10-50 ft or 
more. The largest modern gratings, used in their highest orders, show resolving 
power X/5X in excess of 900,000 in the green region of the spectrum, and in excess of 
1.5 X 10 6 at shorter wavelength. Here X is the mean wavelength of two closely spaced, 
just-resolvable spectral lines and 8X is their wavelength difference. 


314 


OPTICAL COMPONENTS 


References 

1. "Data Sheets,” Eastman Kodak Co., Rochester, N. Y. (1962). 

2. "Data Sheets,” Servo Corp. of America, Hicksville, N. Y. (1962). 

3. "Advanced Optics Engineering, Design and Fabrication,” Bulletin 6-010, Barnes Engineering 
Corp., Stamford, Conn. 

4. "Near-Infrared Transmission Filters,” Progress Report No. 3, Bausch & Lomb Optical Co., 
Rochester, N. Y. (1958). 

5. C. F. Mooney and A. F. Turner, "Infrared Transmitting Interference Filters, Proceedings of 
the Conference on Infrared Optical Materials, Filters, and Films,” Engineer Research and 
Development Laboratories, Fort Belvoir, Va. (1955). 

6. G. Hass and A. F. Turner, Coatings for Infrared Optics, Reprint from Wissenschaftliche 
Verlagsgesellschaft m.b.h. Stuttgart, 143-163. 

7. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detection and Measurement of Infrared 
Radiation, Oxford University Press (1957). 

8. Engineering Notebook, Astrionics Division, Aerojet-General Corp., Azusa, Calif. (1961). 

9. E. D. McAlister, "The Christiansen Light Filter; Its Advantages and Limitations,” Smithsonian 
Miscellaneous Collections, 93, 7 (1936). 

10. R. B. Barnes and T. W. Bonner, Phys. Rev., 49, 732 (1936). 

11. J. Strong, Concepts of Classical Optics, W. H. Freeman and Co., San Francisco (1958). 

12. "Data Sheets,” Infrared Industries, Waltham, Mass. (1962). 

13. N. M. Mohler and J. R. Loofbourow, "Optical Filters,” Am. J. Phys., 20, 579-588 (1952). 

14. J. H. Shenk et al., J. Opt. Soc. Am., 36, 10, 569 (1946). 

15. W. L. Wolfe and S. S. Ballard, "Optical Materials, Films, and Filters for Infrared Instrumen¬ 
tation,” Proc. I.RJ£., 47, 9 (1959). 

16. R. H. Bube, Photoconductivity of Solids, Wiley, N. Y. (1960). 

17. M. R. Holter, S. Nudelman, G. H. Suits, W. L. Wolfe, and G. J. Zissis, Fundamentals of Infrared 
Technology, Macmillan, N. Y. (1962). 

18. F. A. Jenkins and H. E. White, Fundamentals of Optics, 2nd ed. McGraw-Hill, N. Y. (1950). 

19. H. L. Hackforth, Infrared Radiation, McGraw-Hill, N. Y. (1960). 

20. V. Z. Williams, " Infrared Instrumentation Techniques,” Rev. Sci. Instr., 19, 135-178 (1948). 

21. G. L. Clark, The Encyclopedia of Spectroscopy, Reinhold, N. Y. (1960). 

22. H. Ebert, Wied. Ann., 38, 489 (1889). 

23. W. G. Fastie, J. Opt. Soc. Am., 42, 9, 641 (1952). 

24. H. A. Rowland, Phil. Mag., 16, 197 and 210 (1883). 

25. C. R. Runge and F. Paschen, Abhandl. Akad. Wiss. Berlin, Anhang 1 (1902). 

26. W. DeW. Abney, Phil. Trans., 177, II, 457 (1886). 

27. A. Eagle, Astrophys. J., 31, 120 (1910). 

28. R. A. Sawyer, Experimental Spectroscopy, Prentice-Hall, N. Y. (1944). 

29. F. L. O. Wadsworth, Astrophys. J., 3, 54 (1896). 

30. M. Leya, Sci. of Light, 2, 8, Tokyo (1952). 

31. J. B. Hoag, Astrophys. J., 66, 225 (1927). 

32. B. Edlen, Nova Acta Regiae Soc. Sci. Upsaliensis, 9, 6 (1934). 

33. F. Tyren, Z. Physik., Ill, 314 (1938). 


Chapter 8 

OPTICAL MATERIALS 


William L. Wolfe 

The University of Michigan 


CONTENTS 


8.1. Introduction. 317 

8.2. Types of Materials. 317 

8.2.1. Glasses. 317 

8.2.2. Semiconductor Crystals. 324 

8.2.3. Plastics. 325 

8.2.4. Some Recent Hot-Pressed Samples. 326 

8.3. Comparisons of Material Properties. 326 

8.3.1. Transmission Region. 326 

8.3.2. Refractive Index. 328 

8.3.3. Dispersion. 328 

8.3.4. Dielectric Constant. 329 

8.3.5. Melting Temperature. 330 

8.3.6. Thermal Conductivity. 331 

8.3.7. Thermal Expansion. 331 

8.3.8. Specific Heat. 333 

8.3.9. Hardness. 333 

8.3.10. Solubility. 333 

8.3.11. Young’s Modulus. 333 

8.4. Material Data Useful for Lens Design. 335 

8.4.1. Dispersion Equations for Individual Materials. 335 

8.4.2. Herzberger Dispersion Equation. 337 

8.4.3. Refractive Index, Dispersion Values. 338 

8.4.4. G Sums. 341 

8.5. Useful Equations. 347 

8.5.1. Reflection and Transmission at a Single Surface. 347 

8.5.2. Reflection and Transmission by a Single Layer. 348 

8.5.3. Emissivity, Transmissivity, and Reflectivity 

for Partially Transparent Bodies. 349 

8.5.4. Loss Tangent. 350 

8.5.5. Extinction Coefficient. 351 


315 

































OPTICAL MATERIALS 


8.6. Optical Surface Coatings. 351 

8.6.1. Reflective Coatings. 351 

8.6.2. Filter Mirrors. 352 

8.6.3. Protective Coatings. 354 

8.6.4. Antireflection Coatings. 355 

8.6.5. Replica Mirrors. 355 

8.7. Radiation Damage. 358 

8.8. Infrared Transmission of Cooled Optical Materials. 359 

8.9. Optical Properties of Blacks. 359 

8.10. Optical Properties of Water. 362 

8.11. Effects of Space Radiation on Optical Materials. 364 


316 













8. Optical Materials 


8.1. Introduction 

Data on optical, mechanical, thermal, and chemical properties of infrared optical 
materials are presented here. For additional information see [11, publications of 
the various manufacturers of optical materials, and the references. 

8.2. Types of Materials 

Optical materials in current use include glasses, crystals, plastics, and hot-pressed 
materials. The crystals, natural and artificial, are dielectric and semiconducting. 

8.2.1. Glasses. Most optical glasses transmit into the infrared only to a wave¬ 
length of about 2.7 fx and then fairly strong absorption sets in; beyond about 4 fx, glasses 
have no appreciable transmission. Figure 8-1 shows the transmission of several 
infrared-transmitting glasses that do not demonstrate a strong "water-band” absorp¬ 
tion at 2.8 /x but exhibit a slower dropoff in transmission in the 3-g to 4-/x region. 
Figure 8-2 is the standard n-v curve for ordinary glasses, that is, the refraction for 
the D-line plotted against v = nn-\Kriy — nc)- 

8.2.1.1. High-Silica Glasses and Fused Quartz. The names fused quartz, silica 
glass, fused silica, etc., now mean about the same thing: amorphous mixtures that are 
almost entirely silica. Originally the names differentiated among natural and syn¬ 
thetic materials. Some special silicate glasses recover transmission beyond the 2.8-/x 



317 






















318 


OPTICAL MATERIALS 



Fig. 8-2. Dispersion region for standard glasses. 



Fig. 8-3. The transmission of some high-silica glasses (from company data sheets). 

absorption band and give reduced transmission out to about 5 /a. This performance 
is illustrated by the transmission curve in Fig. 8-1 for Corning No. 186S J, which is a 
lead silicate glass. The properties of fused and crystal quartz are given in [1]. Ex¬ 
amples of the properties of a number of commercial varieties (obtained from manu¬ 
facturers’ brochures and private communications) are given below to provide useful 
data and illustrate the small variation in most of the physical constants among the 
glasses and the drastic difference from these in the properties of fused quartz. 

Most of the useful physical properties are listed in Table 8-1. Variations in glasses 
are obtained by use of different purifications and by different annealing procedures. 
The transmission of the samples from the principal suppliers (Corning, Dynasil, Engel¬ 
hard, General Electric, Heraeus, and Sawyer) are given in Fig. 8-3. The 2-mm Vycor 
sample is from Corning; Infrasil is from Engelhard; GE 103, 105, and 106 are from the 
General Electric Lamp Glass Department (the GE 102 and 104 have in addition to 












Table 8-1. Properties of Silica Glasses 


TYPES OF MATERIALS 


319 


o 
• ** 

HO 

O 

• *o 


3 

3 

co 

3 

o 

O 


o 


00 

CO 


CO 


03 


to 

4* 




o 


3- 


CO 

3 

3 

"3 

I 


& 

• o. 

*ao 

• o 

ft5 


cn 

tX 


Oi 

T}< 


CO 

3 

3 

"3 

I 


£ 

e. ^ • cH 

^ CO 

° §-3 


aj 


CO 

l> 


^CO «§ 
tJX) 

3 3 
3 "3 
k° J2 


co 

cx 

X 

o 


CO 

03 


tJ* 

d 


o 


co 

d 


3 

s 

Ss- 

<i> 

-3 


3 i 

•2 r 
£ S 

§ X 

R* i 

O 

KJ r*. 


GO 

o 


1C 

to 


io 

io 

d 


uo 

LO 


t> 

CO 

d 


•w 1 v " 

>-«o * ^ S' 'T“ N 

| •■§ H o 

poo 
o -3 7 
g 3 O 

3-1 © r-l 

O ^ 


o 

a> 

CO 

s 

o 


CO 

co 


CO 

CO 


CO 

CO 


CO 

CO 


o 

(M 


o 

<3. 

'3 o 

3, ^ 

co 


LO 

lO 

<M 

o 


oo 

oo 


3 

•*o 

CO 


£ u 

o 

Ph W 


¥ £ 

^ ,—s 

g £ u 

/*s ft) O 

CO 


'K 3 

^ bo 


Ho 

I 


o 

03 


<N 

3- 


oo 

o 



i-H 


o 

t> 


o 

CO 


LO 

CO 


t—H 

r— t 


oo 

o 


rH 

<N 


<N 

oq 



CO 

i-H 

f-> LO 

o 

o 

O O 

»-H 



w 

W 

> ^ 

o 

O 


03 

t> 

O 


I> 

co 

CO 


o 

<M 

c4 


CO 

o 


<M LO 

o o 


o 

o 

3> 


<N 

d 


T3 

H 

cs 

Xl CC 
*2 
g 5 c 
w ^ 


O CD LO 

t> I t> 
Tt< S U0 


LO 

CO 


<N 


N 


t, 

to « 
>> 3 
>* nf 
o 


10 Me 


Table 8-2. Properties of Silicate, Germanate, and Aluminate Glasses 


320 


OPTICAL MATERIALS 


CO O) 

§ 

^.oS 

pi 


o o 
o o 

o o 


in 10 w 
o o o 
o o o 

odd 


o •«. 
C 3 
is o 

o ■** 
<u » 

s 
o 
O 


<*> 

• -- 

Q 


o 


CO t> 


05 05 05 


05 

05 




oo o 

O' CD 
Tf 


CO CM 
05 t> Tf 

m w m 


CO 

3 

— 

3 

13 

I 


- « 

05 


I I I 


co 

CO 

CM 


CO 

3 

3 

13 

s 


S! 

© ft, &, 
3 d 1 


O 

CM 

W 

in 


CO 


CO 

t>C -2 
3 3 
3 "3 
O <3 

* ^ 


CO 

CU 


C— rH 
00 05 


in 

in 


CM CM H 

in 6 m 


co 

in 


c 

1 

<51 

-3 


3 T 

■2 

g u 

r 


o 


co t> 

00 05 


"tf 

00 


H CD ^ 
00 05 00 




3 

2 

cu 

■3 


X ^ 

o 

cC U 

° o 

45 

3 ° _ 

~ - s 

o 


<J - 

^ -o " 


o 

O 


I I 


CM 

CO 


■K» 

3 

£ 

o 

co 


tj* t> 

d o 


•S 

5 gO 

■h <5» w 

co 


05 oo 
CM 00 
CO CO 


I I I 


.§pa 

3 1*6 
£ 

o-^ 

CO 


m o 

oo H 

t> co 


co 

co 

oo 


o o o 
o o co 

05 00 t- 


o 

■'t 

o 

CM 







00 

00 

• ^ (J 

i-H in 


t> 


co o 

in o 


© 

m 

i—t 

05 

^ tc 

a “ 

^ co 

•M* 

CO 

■<* 

in 

CO 


3 

<55 

-K 4 

3 


co 

45 

03 

C 

CO 

£ 

u 

45 

O 


30 

C 

• 

C 

u 

O 

O 


CM O 

m co 

C— i—I 
05 O 


03 

45 

ce 


GO 


00 

in 

rH 

CO 

PQ 

£ 


03 

45 

a 

c 

£ 

3 


JO 
£ 

=3 
-c 2 
S 

3 
3 02 
00 


CM 


O 

CM CM 


05 


05 

HH 

05 


05 

M 

05 


W 

o 


x 

o 

13 

o 

3 


Barr & Stroud 
BS 37B 


TYPES OF MATERIALS 


321 


these curves a narrow, deep 2.8 -p absorption); Suprasil is the best of the Heraeus 
fused silicas; the Sawyer material is virtually identical to Infrasil; and the Dynasil is 
similar to Infrasil but with a deep absorption at 2.8 p. Sometimes this absorption can 
be alleviated by careful attention to water elimination during preparation. 

8.2.1.2. Silicate Glasses. The National Bureau of Standards in its studies of in¬ 
frared-transmitting glasses has produced some silicates with fairly good transmission 
beyond the OH band at 2.8 p. One example is given in Table 8-2. Actually this is 
a composite set of properties (taken from the NBS report series) and indicative of 
no particular glass. The properties of individual samples will approximate these 
values but change with variation in composition. 

8.2.1.3. Germanate Glasses. The properties of several varieties of germanate glasses 
are displayed in Tables 8-2 and 8-3 and Figs. 8-4 and 8-5. The properties of individual 
samples will approximate the values given but change with variation in composition. 

8.2.1.4. Calcium Aluminate Glasses. Bausch and Lomb RIR-2, -10, -11, -12, and 
-20, GE Lucalox, and Barr and Stroud 36A (old type) and 37B are all types of calcium 
aluminate glass. Properties are given in Tables 8-2 and 8-4 and Fig. 8-6. The 2.8 -p 
absorption band can be eliminated by careful attention to humidity during preparation, 
e.g., by vacuum melting. 

8.2.1.5. Nonoxide Glasses. Arsenic-modified selenium glass, arsenic-sulfur glass, 
and other nonoxide glasses are not restricted to those containing the silicate or alumi¬ 
nate radical. They are described in [1]. Some newer materials include mixtures of 
Se, Te, S, As, Ge, and a few other heavy atoms. Reports on these materials can be 
found in [2] and [3]. 


Table 8-3. Refractive Index of 
National Bureau of Standards F998 


Wavelength 

Germanate Glasses 

(p) 

F998 

0.4358 

1.88997 

0.4861 

1.87470 

0.5461 

1.86242 

0.5780 

1.85755 

0.5893 

1.85597 

0.6439 

1.84986 

0.6563 

1.84866 

0.8521 

1.83632 

1.0140 

1.83082 

1.1287 

1.82801 

1.3622 

1.82374 

1.5295 

1.82142 

1.6606 

— 

1.6932 

1.81930 

1.7012 

— 


Wavelength Germanate Glasses 

(p) F998 

1.9701 

2.1526 1.81364 

2.2493 

2.3126 1.81151 

2.4374 1.81000 

2.5947 1.80797 

2.6685 1.80680 

2.998 

3.3033 1.79706 

3.422 1.79490 

3.5078 1.79345 

4.225 

4.253 1.77863 

4.281 

5.138 1.75437 

5.343 1.74805 


322 


OPTICAL MATERIALS 



WAVELENGTH (p) 

Fig. 8-4. Transmission of several germanate glasses 
(from company data sheets and NBS reports). 



Fig. 8-5. Index of refraction of Corning 
No. 9752 glass. 






TYPES OF MATERIALS 323 

Table 8-4. Refractive Index of Bausch and Lomb Calcium Aluminate Glasses 


Wavelength 

(g) 

RIR-10 

RIR-11 

RIR-12 

RIR-2 

RIR-20 

0.4047 

_ 

_ 

__ 

1.82800 

1.91449 

0.4341 

— 

— 

— 

1.81806 

1.90155 

0.4359 

— 

— 

— 

1.81746 

1.90082 

0.4861 

1.66057 

1.67887 

1.66647 

1.80536 

1.88529 

0.5461 

1.65385 

1.67109 

1.65919 

1.79558 

1.87274 

0.5876 

— 

— 

— 

1.79060 

1.86639 

0.5893 

1.65022 

1.66699 

1.65532 

1.79041 

1.86616 

0.6563 

1.64588 

1.66239 

1.65085 

1.78443 

1.85866 

1.0140 

1.6352 

1.6506 

1.6397 

1.76988 

1.84044 

1.1287 

1.6334 

1.6486 

1.6378 

1.76741 

1.83762 

1.3620 

1.6304 

1.6455 

1.6346 

1.76343 

1.83333 

1.5295 

1.6285 

1.6435 

1.6328 

1.76104 

1.83000 

1.6606 

1.6271 

1.6420 

1.6313 

1.75920 

1.82909 

1.8131 

1.6255 

1.6404 

1.6297 

1.75718 

1.82722 

1.9701 

1.6238 

1.6386 

1.6280 

1.75503 

1.82527 

2.1526 

1.6216 

1.6364 

1.6259 

1.75238 

1.82290 

2.2493 

— 

— 

— 

1.75103 

1.82183 

2.3254 

1.6196 

1.6344 

1.6239 

1.74984 

1.82073 

2.4374 

1.6182 

1.6329 

1.6224 

1.74806 

1.81924 

2.577 

— 

1.6310 

1.6206 

1.74582 

1.81732 



Fig. 8-6. Transmission of several calcium aluminate glasses; thickness, 
2 mm (from company data sheets and NBS reports). 






324 


OPTICAL MATERIALS 


8.2.2. Semiconductor Crystals. The cuton wavelength and the transmittance of 
semiconductors are functions of temperature (Fig. 8-7) and purity (Fig. 8-8). Figure 8-9 
shows the transmission of various semiconducting materials. 

By their basic nature semiconducting materials have small energy gaps which 
correspond to cuton wavelengths in the infrared. Increasing the temperature of the 
material in effect narrows the gap, thereby increasing the cuton wavelength. It also 
increases the probability that electrons can have energies characteristic of the conduc¬ 
tion band. Many additional data will be found in [1] and in the several journals on 
solid-state physics. 



Fig. 8-7. Effect of temperature on semiconductor 
transmission; sample is 1.17 mm, 30 ft cm Ge. 



Fig. 8-8. Effect of purity on semiconductor 
transmission; samples are 0.2 mm, 7 x 10 -3 
O cm InSb and 0.2 mm, 2.5 x 10~ 4 ft cm InSb. 



Fig. 8-9. Transmission of selected semiconductor materials. 









































TYPES OF MATERIALS 


325 



Fig. 8-10. Transmission of polyethylene; 
thickness, 0.1 mm. 



Fig. 8-11. Transmission of Plexiglas; 
thickness, 0.2 mm. 


8.2.3. Plastics. Among the common plastics used for infrared applications are 
polyethylene and polymethylmethacrylate (available commercially as Lucite or Plexi¬ 
glas respectively). Transmission curves of these two materials are shown in Figs. 8-10 
and 8-11. These curves are representative of the transmission of thin films of many 
different plastics. Except for narrow bands, where only a small amount of energy is 
absorbed, and for broader bands in some materials, the transmission is relatively good. 
For thicker samples, however, the regions of small absorption deepen rapidly and widen 
considerably, and the absorption becomes so great that the material may no longer 
be satisfactory for the intended use. Table 8-5 shows the wavelengths at which various 
molecules and molecular groups absorb infrared radiation. These wavelengths are 
the characteristic absorption bands of the bonds or groups noted. 


Table 8-5. Absorption Wavelengths 
of Characteristic Groups [1] 


Molecular Group 

O —H 

N-H 

C-H 

Carbonyl 

Methyl 

Ester 

C-0 

C-Cl 

Si-0 

Al-0 

Ge-0 


Wavelengths of 
Absorption 
(/a) 

2.8 

3.03, 6.12, 6.46 

3.4, 6.81 
5.75 

7.26, 7.71 
8, 9.74 

9(broad), 13.47-14.20 
14-15 

4.5, 9.5 
— 6.5 

-5.5, 11.6 

















326 


OPTICAL MATERIALS 


Kel-F, a polymer of trifluorochloroethylene, is used for windows and coatings although 
in thicknesses greater than about 0.25 in. it is difficult to manufacture with suitably 
high transmission. It has a low thermal conductivity, and its refractive index may 
vary throughout a sample. The transmittances of two different thicknesses of Kel-F 
are shown in Fig. 8-12. 



8.2.4. Some Recent Hot-Pressed Samples. Data are now becoming available 
on various hot-pressed materials. Irtran-1 and -2, Eastman Kodak products, are 
described in the supplement of [2]. Irtran-3, another Kodak product, is pressed CaF 2 ; 
it has basically the same transmission and refractive index as the single crystal. 
Irtran-4, pressed ZnSe, transmits to about 20 p, (0.04-in. sample transmits 50% at 
21 p) with a dip at 9 p. Harshaw has just announced T-12, a milky white sample that 
is apparently hot pressed. Few data are available on its composition and properties. 

8.3. Comparisons of Material Properties 

8.3.1. Transmission Region. Figure 8-13 shows the transmission regions of most 
of the infrared optical materials. The white bars represent the wavelength region 
in which a particular material transmits appreciably. 

The limiting wavelengths, for both high and low cutoff, are arbitrarily chosen as 
that wavelength at which a sample of 2-mm thickness has 10% external transmittance. 
For some cases this criterion is insufficient, such as in the consideration of semicon¬ 
ducting materials where purity and temperature must also be specified. Some semi¬ 
conductors violate the criterion in another way: materials such as indium antimonide 
have an external transmittance of less than 10% for a 2-mm-thick sample even in 
their most transparent regions; these are indicated by an asterisk (*). 

Several different "endings” are used for the bars of the chart in Fig. 8-13. Each 
has a specific meaning: a bar with a straight vertical ending indicates that the cutoff 
exists at the wavelength represented by the end of the bar exactly as defined above; 
a bar with an S-shape ending represents a material which cuts off at approximately 
that wavelength; a bar ending in an angle indicates that the material transmits at least 
to that wavelength, and probably further. Measurements made on materials in this 
last group have not been made to sufficiently long or sufficiently short wavelengths 
to determine the cutoff. 








COMPARISONS OF MATERIAL PROPERTIES 


327 


.2 

X 


.3 

nz 


.4 .5 .6 .7 .8 .9 1.0 

m. 


2.0 
~T ' 


3.0 4.0 5.0 


10.0 


20.0 30.0 


100.0 micron 


(6.125 


A DP 


. 7 .. ii. 'rir 


B 


■i. .1 ..l. L 1 ,t: 


Xz? 




T \ T FT 


X!? 


Vd.12, , c 

:ryst/> 

lL ouartz 

1" '} ' 7 "l 1 r* V r 1 " ’ ■ 

.— ■■XC 


SO. 12 I 

'USED 

L—^ -- ■ h. 1. I.-.-.. 

SILICA 

—.. 

... 


Cl 


iO.4 


US GALLIUM PHOSPHIDE* 4.51 

.. I ■ ' ' f. ■ ' .1 ■ 


V6.2 CALC1TE 


CALCIUM ALUMINATE GLASS 575] 

XX ' T l ' f.. .1 . ' , i, ,1 if 


B 


TCLil' 


0.6 SPINEL 

" 1 1 1 '1-- 


TCT 


SO. 4 3 RUTILE 


x 


- .. ■ - .t «••••■<. ) t <■ ■' ■■ ■' T~ 

JO.14 SAPPHIRE 

(0.39 STRONT UM TITANATE 


'I " .'C 




' i . .'i.' i. >. 


x 


c 


INDIUM ARSENIDE [378 
LEAD SULFIDE (film)* [37(5 


0.5 


t:4 

LEAD SELENIDE (film)* 

LEAD TELLURIDE (film)* 

l.-l-.. ■■■-■ ■ t ■■■■ i 


16.11 MAGNESIUM FLUORIDE 


BARIUM TITANATE 

" 1 





fo725 


MAGNESIUM OXIDE 

T. . 


TELLURIUM C5 

■■ . .i . .. .. - w 


X7X 


f."l‘ 


JO. 12 LITHIUM FL 


yORIDE 


H'1 


fo.l'3" CALCIUM FLUORIDE 


■i . k i: 


9.0 




SO. 2 5 EAR 


<0.19 SODIUM 


JM 


m 




___ _ t . t . 

ARSENIC TRISULFIDE GLASS ~ ~ ' ' ' 13f 


1.0 


1.0 


INDIUM PHOSPHIDE 

-3- 




1.0 


LR TRAN-2 

m 




Exun 


h! 


£1 


GALLIUM ARSENIDE 

-1- 


•}. ..ixt •..i. i'.vrr 


J4# 


fiSL 


SILICON 


rrrr::i . r 




3ZZE3ZnUZE 


ID 


■I-■■ I- t -t.I f 


H? 


10.21 POTASSIUM CHLORIDE 




30 

~T + l.' 't ' 1 l'i' “ 

^..1— 


. . , .r.' .r. i .T. iv.f. 1 '. 



(o'.25 LEAD FLUORIDE 


FLUORIDE 

' J ' : 1 ' v ' ■ r i i. v rrr 

INDIUM ANTIMONIDE* |7 5 




M> 


l'. .'. . 1 .. r. .{ 'Kf. f. i: 


m 






:r-:r:nvi7T 


IB 


0.9 CADMIUM TELLURIDE 16? 

^ic.:_:.:.r. :. ;xr;. cr ..x .v.x:o:-i , .i , ,:ccc.,. 

0.8 ARSENIC MODIFIED SELENIUM GLASS 18> 


3ZZECE3HIO: 


1 0.21 SODIUM CHLORIDE 


1.0 AMORPHOUS SELENIUM 

”—’— txxt/sttxi. 3.. i '.i.r-rr 

Ill GERMANIUM 


20 


& 


E 




SILVER CHLORIDE 

T-’rT’I. - 




.■f.'.T nrr 


26] 


3ZX 


28 { 


TOT 


KRS-6 

40742 THALLIUM CHLORIDE 


■I" TTri'T 


Sti.'Ss" " p6tasVium bromide 




:f ., . i . . t. , t r.t.M 




VOs 




. ..... t'J.t t 

10.42 T 


T 


x 


-I" 1 ■IM't'T 


40 


T7TTT 


s 


POTASSIUM IODIDE 


HALLIUM BROMIDE 


.1.'. ..T',.t'J..L1. 


i i tT 


. 40} 

40? 

' 1.f ' '7 


1T3.X1I, 


CESIUM BROMIDE 


■ f.i. . r ' 1 ". i"T ' . r .r ~r 


ZL 


45? 

■1' ■' .1 


x.xn.X-i-T 


(05 CESIUM IODIDE 

_ "" * I ■ ' < ' - 

SO, 2 5 DIAMOND 




X 




-r—r-P 


{' I T f 


1 .'.'. 1 . .'t' .I.'i'.rn. 


x 


\v. 


80? 

f> 


'Maximum external transmittance of less than 10% 


1961 


Fig. 8-13. Transmission regions of optical materials, 2 mm thickness; cutoff is defined 
as 10 percent external transmittance, and materials marked with an asterisk never have 
external transmittance as high as 10 percent. 
















































































































































































































































328 


OPTICAL MATERIALS 


8.3.2. Refractive Index. Figure 8-14 presents curves of refractive index versus 
wavelength and illustrates the range of the refractive indices of many of the materials. 
For crystals whose refractive index varies with direction, only the refractive index cor¬ 
responding to the ordinary ray has been plotted. The refractive indices of tellurium 
for both the ordinary and extraordinary rays have been omitted because of their ex¬ 
tremely high values (approximately 6.237 and 4.789, respectively at 12 /a). 


4.0 


X 3.0 
w 
Q 
Z 

« 

> 

H 

U 

t 

fa 

« 2.0 


1.0 

0.1 1.0 10 50 

WAVELENGTH (fi) 

Fig. 8-14. Refractive index values. 


8.3.3. Dispersion. The data of Fig. 8-14 are plotted in Fig. 8-15 to show the rate 
of change of the refractive index versus wavelength. 




o.oooi 

1.0 

WAVELENGTH (u) 

Fig. 8-15. dn/d\ versus X for selected materials. 



















COMPARISONS OF MATERIAL PROPERTIES 


329 


The dispersion in the desired wavelength interval can be determined from Figs. 8-14 
and 8-15. If greater accuracy is required, the dispersion can be calculated from the 
refractive index information contained in this chapter. 

8.3.4. Dielectric Constant. Values for the dielectric constant as a function of 
frequency and temperature are given in Table 8-6. These values are the relative 
dielectric constants of materials, that is, the ratios of the dielectric constants of the 
material to that of a vacuum. They include measurements taken at microwave fre¬ 
quencies and indicate such peculiarities as variation with orientation. Since dielectric 
properties depend upon purity, particularly in semiconductors, the purity of the sample 
measured is given, where available. 

When the dielectric constant is measured with the electric field parallel to the c axis 
(the optic axis), the measurement is identified with a superscript p\ when the electric 
field is perpendicular to the optic axis, the measurement is identified with a superscript s. 


Table 8-6. Dielectric Constant of Optical Materials [1] 


Material 

Dielectric 

Frequency 

Temperature 

Remarks 

Constant 

(cps) 

(°C) 


Fused silica (Si0 2 ) 

3.78 

10 2 to 10 10 

25 


Silica glass 

3.81 

10 8 

20 


Crystal quartz (Si0 2 ) 

4.27" 

10 7 

17 to 22 



4.34 s 

10 7 

17 to 22 


Potassium chloride (KC1) 

4.64 

10 6 

29.5 


Potassium bromide (KBr) 

4.90 

10 2 to 10 10 

25 


Potassium iodide (KI) 

4.94 

10 6 

— 


Cesium iodide (Csl) 

5.65 

10 6 

25 


Sodium chloride (NaCl) 

5.90 

10 2 to 10‘° 

25 


Amorphous selenium (Se) 

6.00 

10 2 to 10 10 

25 


Sodium fluoride (NaF) 

6.0 

10 6 

19 


Selenium crystal (Se) 

6.0 

10 2 to 10 10 

— 


Cesium bromide (CsBr) 

6.51 

10« 

25 


Calcium fluoride (CaF 2 ) 

6.76 

10 5 

— 


Sodium nitrate (NaNO :! ) 

6.85 

10 s 

19 


Mica, glass bonded, injection 

6.9 to 9.2 

10 6 

Room 


Barium fluoride (BaF 2 ) 

7.33 

10« 

- 


Calcite (CaC0 3 ) 

8.5* 

10 4 

17 to 22 



8.0" 

10 4 

17 to 22 


Sapphire (A1 2 0 3 ) 

10.55" 

10 2 to 10 8 

25 



8.6* 

10 2 to 10 10 

25 


Arsenic trisulfide glass (As 2 S 3 ) 

8.1 

10 3 to 10 s 

- 


Spinel (Mg0-3.5A1 2 0 3 ) 

8 to 9 

- 

- 


Lithium fluoride (LiF) 

9.00 

10 2 to 10 10 

25 


Magnesium oxide (MgO) 

9.65 

10 2 to 10 8 

25 


Cadmium telluride (CdTe) 

11.0 

10 3 to 10 s 

— 

5.5 X 10 13 carriers/cc 

Silicon (Si) 

13 

10 10 

- 


Silver chloride (AgCl) 

12.3 

— 

Room 


Germanium (Ge) 

16.6 

10 10 

- 

9.0 ohm-cm resistivity 

Lead sulfide (PbS) 

17.9 

10 6 

15 


Thallium bromide (TIBr) 

30.3 

10 3 to 10 7 

25 


Thallium bromide-iodide (KRS-5) 

32.5 

10 2 to 10 7 

25 


Thallium chloride (T1C1) 

31.9 

10 6 

- 


Thallium bromide-chloride (KRS-6) 

32 

10 2 to 10 5 

25 


Potassium dihydrogen phosphate (KDP) 

44.5 to 44.3* 

10 2 to 10 8 

— 



21.4 to 20.2" 

10 2 to 10 8 

- 


Ammonium dihydrogen phosphate (ADP) 

56.4 to 55.9* 

10 2 to 10 8 

- 



16.4 to 13.7" 

10 2 to 10 10 

- 


Titanium dioxide (Ti0 2 ) 

170" 

10 4 to 10 7 

25 



86* 

10 2 to 10 7 

25 


Strontium titanate (SrTiO :i ) 

234 

10 2 to 10 10 

25 


Barium titanate (BaTi0 3 ) 

1240 to 1200 

10 2 to 10 8 

25 



"Dielectric constant measured parallel to c axis. 

* Dielectric constant measured perpendicular to c axis. 
— Value not indicated. 


OPTICAL MATERIALS 


330 


8.3.5. Melting Temperature. The melting temperature of optical materials (or 
the softening temperature if appropriate, or for glasses) is given in Table 8-7. 

Table 8-7. Melting, or Softening, Temperature 
of Optical Materials [1] 


Material 

Temperature 

(°C) 

Amorphous selenium (Se) 

Arsenic modified selenium glass [Se(As)l 
Arsenic trisulfide glass (AS 2 S 3 ) 

Potassium dihydrogen phosphate (KDP) 
Sodium nitrate (NaN0 3 ) 

Gallium arsenide (GaAs) 

Thallium bromide-iodide (KRS-5) 

Thallium bromide-chloride (KRS-6) 

Thallium chloride (T1C1) 

Tellurium (Te) 

Silver chloride (AgCl) 

Thallium bromide (TIBr) 

Gallium phosphide (GaP) 

Indium antimonide (InSb) 

Cesium iodide (Csl) 

Cesium bromide (CsBr) 

Gallium antimonide (GaSb) 

Potassium iodide (KI) 

Potassium bromide (KBr) 

Potassium chloride (KC1) 

Sodium chloride (NaCl) 

Borosilicate crown glass 

Lead fluoride (PbF 2 ) 

Lithium fluoride (LiF) 

Calcite (CaC0 3 ) 

Cadmium sulfide (CdS) 

Lead telluride (PbTe) 

Germanium (Ge) 

Indium arsenide (InAs) 

Sodium fluoride (NaF) 

Cadmium telluride (CdTe) 

Indium phosphide (InP) 

Lead selenide (PbSe) 

Lead sulfide (PbS) 

Gallium arsenide (GaAs) 

Barium fluoride (BaF 2 ) 

Calcium fluoride (CaF 2 ) 

Silicon (Si) 

Crystal quartz (Si0 2 ) 

Barium titanate (BaTi0 3 ) 

Fused silica (Si0 2 ) 

Titanium dioxide (Ti0 2 ) 

Sapphire (A1 2 0 3 ) 

Spinel (MgO-3.5 A1 2 0 3 ) 

Strontium titanate (SrTi0 3 ) 

Magnesium oxide (MgO) 

35* 

70* 

210* 

252.6 

306.8 

400$ 

414.5 

423.5 

430 

449.7 

457.7 

460 

>500 

523 

621 

636 

720 

723 

730 

776 

801 

820* 

855 

870 

894.4$ 

900t 

917 

936 

942 

980 

-1040 

1050 

1065 

1114 

1238 

1280 

1360 

1420 

<1470 

1600 

-1710 

1825 

2030 

2030 to 2060 

2080 

2800 

*Softening temperature. 
tSublimation temperature. 

^Dissociation temperature. 



COMPARISONS OF MATERIAL PROPERTIES 331 

8.3.6. Thermal Conductivity. Values of thermal conductivity for optical materials 
are given in Table 8-8. For crystals that exhibit anisotropy, the orientation of the 
heat flow with respect to the c axis is noted; values are given for the heat flow parallel 
(p) and perpendicular (s) to the c axis. 

Table 8-8. Thermal Conductivity of Optical Materials [1] 



Thermal 

Temperature 

(°C) 

Material 

Conductivity 
[KL 4 cal/(cm sec C°)] 

Diatomaceous earth 

1.3 

"ordinary” 

Arsenic modified selenium glass [Se(As)] 

3.3 

_ 

Arsenic trisulfide glass (As 2 S 3 ) 

4.0 

40 

Thallium bromide-iodide (KRS-5) 

13 

20 

Thallium bromide (TIBr) 

14 

43 

Lead sulfide (PbS) 

16 

_ 

Thallium bromide-chloride (KRS-6) 

17.1 

56 

Thallium chloride (T1C1) 

18 

38 

Ammonium dihydrogen phosphate (ADP) 

17 p 

42 


30* 

40 

Cesium bromide (CsBr) 

23 

25 

Cesium iodide (Csl) 

27 

25 

Silver chloride (AgCl) 

27.5 

22 

Fused silica (Si0 2 ) 

28.2 

41 

Potassium dihydrogen phosphate (KDP) 

29 p 

39 


32* 

46 

Barium titanate (BaTi0 2 ) 

32 

Room 

Calcite (CaC0 3 ) 

132 p 

0 


111* 

0 

Potassium bromide (KBr) 

115 

46 

Tellurium (Te) 

150 

_ 

Sodium chloride (NaCl) 

155 

16 

Potassium chloride (KC1) 

156 

42 

Crystal quartz (Si0 2 ) 

255 p 

50 


148* 

50 

Calcium fluoride (CaF 2 ) 

232 

36 

Lithium fluoride (LiF) 

270 

41 

Barium fluoride (BaF 2 ) 

280 

13 

Titanium dioxide (Ti0 2 ) 

300 p 

36 


210* 

44 

Spinel (MgO-3.5 A1 2 0 3 ) 

330 

35 

Cadmium sulfide (CdS) 

380 

20 

Sapphire (A1 2 0 3 ) 

600 p 

26 


550* 

23 

Magnesium oxide (MgO) 

600 

20 

Indium antimonide (InSb) 

850 

20 

Germanium (Ge) 

1400 

20 

Silicon (Si) 

3090 

40 

Silver (Ag) 

10060 

18 

p Thermal conductivity measured with heat flow parallel to c axis. 


•Thermal conductivity measured with heat flow perpendicular to c axis. 

— Value not indicated. 



Remarks 


Ceramic material 


n-type, 40 ohm-cm 
resistivity 
P-type 


8.3.7. Thermal Expansion. Table 8-9 shows the linear coefficient of thermal 
expansion of various optical materials. For crystals exhibiting anisotropy, the orienta¬ 
tion of heat flow with respect to the c axis is stated; values are given for heat flow 
parallel (p) and perpendicular (s) to the c axis. 


332 


OPTICAL MATERIALS 


Table 8-9. Linear Coefficient of Thermal Expansion 
of Optical Materials [1] 



Coefficient of 

Average 


Material 

Thermal 

Temperature or 

Remarks 


Expansion 

Temperature Range 



io-«/c° 

(°C) 


Fused silica (Si02) 

0.5 

20 to 900 


Invar 

0.9 

20 


Silicon (Si) 

4.2 

25 


Cadmium sulfide (CdS) 

4.2 

27 to 70 


Cadmium telluride (CdTe) 

4.5 

50 


Indium antimonide (InSb) 

4.9 

20 to 60 


Indium arsenide (InAs) 

5.3 

— 


Germanium (Ge) 

5.5 to 6.1 

25 


Gallium arsenide (GaAs) 

5.7 

— 


Spinel (MgO-3.5 A1 2 0 3 ) 

5.9 

40 


Sapphire (A1 2 0 3 ) 

6.7 P 

50 



5.0 s 

50 


Borosilicate crown glass 

9 

22 to 498 


Titanium dioxide (Ti0 2 ) 

9.19 p 

40 



7.14 s 

40 


Strontium titanate (SrTi0 3 ) 

9.4 

— 


Crystal quartz (Si0 2 ) 

7.97 p 

0 to 80 



13.37 s 

0 to 80 


Sodium nitrate (NaN0 3 ) 

12 p 

50 



11 s 

50 


Magnesium oxide (MgO) 

13.8 

20 to 1000 


Copper (Cu) 

14.09 

-191 to 16 


Tellurium (Te) 

16.75 

40 


Barium titanate (BaTi0 3 ) 

19 

10 to 70 

Ceramic 


6.2 p 

4 to 20 

Single crystal 


15.7 s 

4 to 20 

Single crystal 

Calcium fluoride (CaF 2 ) 

24 

20 to 60 


Arsenic trisulfide glass (As 2 S 3 ) 

24.6 

33 to 165 


Calcite (CaC0 3 ) 

25 p 

0 



-5.8 s 

0 


Silver chloride (AgCl) 

30 

20 to 60 


Amorphous selenium (Se) 

34 

— 

Estimated 

Sodium fluoride (NaF) 

36 

Room 


Potassium chloride (KC1) 

36 

20 to 60 


Lithium fluoride (LiF) 

37 

0 to 100 


Potassium iodide (KI) 

42.6 

40 


Potassium bromide (KBr) 

43 

20 to 60 


Sodium chloride (NaCl) 

44 

—50 to 200 


Cesium bromide (CsBr) 

47.9 

20 to 50 


Thallium bromide-chloride (KRS-6) 

1 50 

20 to 100 


Cesium iodide (Csl) 

50 

25 to 50 


Thallium bromide (TIBr) 

51 

20 to 60 


Thallium chloride (T1C1) 

53 

20 to 60 


Thallium bromide-iodide (KRS-5) 

58 

20 to 100 



p Thermal expansion measured parallel to c axis. 
’Thermal expansion measured perpendicular to c axis. 
— Value not indicated. 


COMPARISONS OF MATERIAL PROPERTIES 


333 


8.3.8. Specific Heat. Table 8-10 lists the specific heat of optical materials. The 
specific heat is given at constant pressure C p rather than at constant volume C„, al¬ 
though the numerical difference is negligible for most purposes. 

8.3.9. Hardness. Values of hardness for several optical materials are given in 
Table 8-11. Knoop values with the indenter aligned in either the (100) or the (110) 
direction are tabulated. The indenter load is given when it is known. Where Knoop 
values are not available, Moh or Vickers values are given. 

8.3.10. Solubility. Values of the water solubility of optical materials at various 
temperatures are given in Table 8-12. 

8.3.11. Young’s Modulus. Values of Young’s modulus for several optical materials 
are given in Table 8-13. The calculated values are obtained from the values of the 
elastic moduli by a method described in [1]. 


Table 8-10. Specific Heat of Optical Materials [1] 


Material 

Thallium bromide (TIBr) 

Tellurium (Te) 

Cesium iodide (Csl) 

Thallium bromide-chloride (KRS-6) 
Lead sulfide (PbS) 

Thallium chloride (T1C1) 

Cesium bromide (CsBr) 

Germanium (Ge) 

Potassium iodide (KI) 

Barium titanate (BaTi0 3 ) 

Silver chloride (AgCl) 

Potassium bromide (KBr) 
Potassium chloride (KC1) 

Silicon (Si) 

Titanium dioxide (Ti0 2 ) 

Sapphire (A1 2 0 3 ) 

Crystal quartz (Si0 2 ) 

Calcite (CaC0 3 ) 

Sodium chloride (NaCl) 

Calcium fluoride (CaF 2 ) 

Magnesium oxide (MgO) 

Fused silica (Si0 2 ) 

Sodium nitrate (NaN0 3 ) 

Sodium fluoride (NaF) 

Lithium fluoride (LiF) 

— Value not indicated. 


Specific Heat 

Temperature 

(°C) 

0.045 

20 

0.0479 

300 

0.048 

20 

0.0482 

20 

0.050 

— 

0.052 

0 

0.063 

20 

0.074 

0 to 100 

0.075 

-3 

0.077 

-98 

0.0848 

0 

0.104 

0 

0.162 

0 

0.168 

25 

0.17 

25 

0.18 

25 

0.188 

12 to 100 

0.203 

0 

0.204 

0 

0.204 

0 

0.209 

0 

0.22 

— 

0.247 

0 

0.26 

0 

0.373 

10 


334 


OPTICAL MATERIALS 


< 


86 

E 

£ 


2 
< 

2 

w 

H 
< 

s s 

3 * 

O ^ be 

hh a 

£ «§ 

JT CO — 


fc 

o 

>* 

Eh 


« 

J 

o 

m 


* 

S3333332233223 
,5.2.S.2,2J2_2.E.2i2,3.2.2.2 

C CCflCCcCflCC 



lO CD lO 
<N <N <N 


r- 


00 O O 00 
H <N <N H 


io o o o o o 

<N (M 


^OOlOlOhr'IN^^ 

30©OprH01COCO<N 

^bddddbdbH* 


r- t> t> 
csi co h« io 
<n co co co 


io 

CO CO 
io t> 


CO IO 

h« 

CN <N 


be 


tf 

^2 



w 

C 

/—N 

n 


Eh 

1 

o 

N 

-< 

£ 

M 

o — 



< 

IO 

‘0? ^ 
® 9 



CO 

N CO 

(N 

'rH 

i 


6 

bo 

s 

t - 

to as 
3 .£ 

O’ — 

GO 


w ' 

— cn 

U 


'aS 

p 

1 'S 

i—] 

CQ 

<; 


‘SL 



co 

O Ck 







S -4 

$ 

as 

* 

be 

o 

o 

r—H 

'So 


c 

c 0 

X 

- 4 -> 

co 

co 

-2 

CO 

3 

o3 

0) 

6 

ju 

3 

3 


2 

< 


co 

H* 

L 


05 



L. — 

w 

^ $9 

Eh 

< 

c ^ 
■§9 


3 O 

c 

o 

HH 

£ 

c 

•2 

o 

L 

Q 

B| 

o 

'u 

CO 


CO 

CO r; 

co c 

w 

"03 ^ 

Z 

Q 

C 

b a. 

ffi 


rH 

rH 

1 

oo 

H 


J 

9 

•2 

1 

H 

1 


8 

A 

*c 

2 

'S 

c 

3 

a 

S 


c 

* 

o 

— 

be 

6 

3 

3 

o 

CS 

> 


'u 'u 

& £ 

3 £ 


6 

3 

z 

A 

O 

s 


3 

z 

J 3 


oooooooooooo 

oooooooooooo 

NNIN<NNU5iniOiONINiN 


OOOOOOOOOOOO 

oooooooooooo 

NU5iONIOiOiO®rtiOWN 


o o o o 

i-HI O rH o 


o o o o o o 
i—i O <—i o »-< o 


i-4 



CO 

i < 

<D 



bo 


o 

cO 


'S 

c 

> 

CO 

Q) 

CO 

CD 

& 

T; 

C 

^2 

V 

Dh 

5 

'E 


o 

o 

50 


o 

o 

m 


u 

03 


© © 
in o 


o o 
rH o 


o o 

O rH 


aoNcoiqaojKKiNiNN 

lOt^^oioiHHNNlOOOOi 


in o> in <n oo <n 
oi oi to 6 05 ri n 


cn 


o o 

rH O 


eo oo 
Os od 00 


05 

be 3 s a s 
as o u to g g 

. 2 8| S|| •§ 
& “ | & * -a *3 «S 

® ^ a> 2 3 

i< Qi 


3 

o 

'S 

s 


o 

o 

o 


£ 

o 

-a 

c 

£ 


m in 


WJ UJ uu U UJ I.N 1—< UU U(J r-1 uj t-N 

HNM'fwco®'oinin®o)05 

?> H H H If m to 


PQ 

£ 

05 

TJ 

E 

S 

-O 

£ 

3 


O 

0 - 


o 

05 

"O 

•c 

o 

2 

o 

£ 

3 

'S 3 

to 

cd 

*j 

o 

Cl, 


Lh 

_ m 

o E 

*P 05 
< TJ 

45 £ 

II 

3 I 

>- A 

05 A 
> CO 

A A 
C/D Eh 


o 

Eh 

05 

13 

•c 

_o 

A 

o 

£ 

3 


o 

z 

as 

z 

5 

cd 

L* 

• ^H 

c 

6 

CO 


CQ 

CO 

O 

a) 

T3 


o 

u 

£ 

£ 

•H 

CO 

<D 

O 


CO 


0 ) 

*C 

o 

2 

u 

i 

T 3 

1 

o 

& 

£ 

e 

2 

2 

Jq 

H 


io 


a> 

"O 

1 

05 

-a 

1 

2 

A 

E 

3 

33 

A 

Eh 


Cl, 

C 8 

CQ 

05 

-a 

•n 

o 

3 

A 

£ 

3 

c 

a 

CQ 


C/D 

M 

CO 

5 

® rs 

2 &H 

JS « 

be O 

4 > T 

"U 05 

«c "2 

II 

.2 6 
s 2 
S3 
i 3 

< O 




03 

t> 

oo 


o o o 

io h 
H H CO 


to 

s 3 


A & 

9 s 

m co 

w +1 

cd ‘43 
= 6 
CO .2 


T 3 

a/ 


a> 

2 

S 

o 

E 

2 

'co 

a) 

c 

bo 


5 l co 

4—1 U 

b-i C/D S 


O 

OT 

N 

3 

3 

O' 


t» 

o 


go" 

Eh 

05 

i3 m 

'3 n 

I O 

3 >-h 

£ S © 

. 2—3 

111 

Eh C/D C/D 


^3 9 

45 OQ 

9 o s 2 

05 9 £ '-S 

•- 05 2 £ 

a -g •£ 5 

a 'o .3 -2 
a — is C 

CO 03 cO 

CO O O CQ 


MATERIAL DATA USEFUL FOR LENS DESIGN 

Table 8-13. Young’s Modulus Value 
for Several Optical Materials [1] 


335 


Young's 

Material Modulus Remarks 

(10 6 psi) 


Cesium iodide (Csl) 

Thallium bromide-iodide (KRS-5) 
Cesium bromide (CsBr) 

Arsenic trisulfide glass (As 2 S 3 ) 
Silver chloride (AgCl) 

Thallium bromide-chloride (KRS-6) 
Potassium bromide (KBr) 

Thallium bromide (TIBr) 

Potassium chloride (KC1) 

Potassium iodide (KI) 

Thallium chloride (T1C1) 

Barium titanate (BaTi0 3 ) 

Sodium chloride (NaCl) 

Indium antimonide (InSb) 

Barium fluoride (BaF 2 ) 

Gallium antimonide (GaSb) 
Lithium fluoride (LiF) 

Calcite (CaC0 3 ) 

Fused silica (Si0 2 ) 

Calcium fluoride (CaF 2 ) 

Crystal quartz (Si0 2 ) 

Germanium (Ge) 

Silicon (Si) 

Magnesium oxide (MgO) 

Sapphire (A1 2 0 3 ) 


0.769 

Measured in flexure 

2.3 

Measured in flexure 

2.3 

Measured in flexure 

2.3 

2.9 

Measured in flexure 

3.0 

Measured in flexure 

3.9 

Measured in flexure 

4.28 

Calculated 

4.30 

Measured in flexure 

4.57 

Calculated 

4.60 

Calculated 

4.90 

Single crystal 

16.50 

Ceramic 

5.80 

Measured in flexure 

6.21 

Calculated 

7.70 

Measured in flexure 

9.19 

Calculated 

9.40 

Measured in flexure 
Minimum value 

10.50 p 

12.80 s 

10.60 

11.0 

Measured in flexure 
Minimum value 

11.1 s 

14.1 p 

14.9 

Calculated 

19.0 

Calculated 

36.1 

Calculated 

50.0 


^Young’s modulus measured parallel to c axis. 
'Young’s modulus measured perpendicular to c axis. 


8.4. Material Data Useful for Lens Design 

Previous sections have summarized the properties of optical materials useful for 
windows, domes, lenses, etc. This section includes equations that are principally 
useful for lens design. Every designer knows that final curvatures, etc. must be 
calculated for the material of an individual batch. Similarly, there are variations 
among crystal samples. The data given below hold for one measured sample and may 
be different for other samples. 

8.4.1. Dispersion Equations for Individual Materials. Unfortunately, not enough 
data are available to compare samples, and some materials have not been of sufficient 
interest to warrant calculation of the constants of a dispersion equation. The available 
ones are listed below. 


336 


OPTICAL MATERIALS 


Arsenic Trisulfide Glass 



-1=i 

i = 1 

K, X 2 


X 2 - X, 2 

i 

X,- 2 

Ki 

1 

0.0225 

1.8983678 

2 

0.0625 

1.9222979 

3 

0.1225 

0.8765134 

4 

0.2025 

0.1188704 

5 

750. 

0.9569903 


Cadmium Sulfide 
Ordinary ray: 


Extraordinary ray: 


Cesium Bromide 


ri 2 


= 5.235 + 


1.819 x 10 7 
X 2 - 1.651 x 10 7 


n 2 = 5.239 + 


2.076 x 10 7 
X 2 - 1.651 x 10 7 


n 2 = 5.640752 - 0.000003338X 2 + 


0.0018612 

A 2 


41110.49 0.0290764 

+ X 2 - 14390.4 + X 2 - 0.024964 


Cesium Iodide 

t%2 - 

i-4 



71 

2 \2 _ X 
1 = 1 

.2 

I 

i 


X, 2 

Ki 

1 

0.34617251 

0.0229567 

0.00052701 

2 

1.0080886 

0.1466 

0.02.49156 

3 

0.28551800 

0.1810 

0.032761 

4 

0.39743178 

0.2120 

0.044944 

5 

3.3605359 

161.0 

25921. 

Fused Silica 

n 2 = 2.978645 + 

0.008777808 

84.06224 


X 2 - 0.010609 

96.00000 - X 2 


Magnesium Oxide 


n 2 = 2.956362 - 0.1062387X 2 - 0.0000204968X 4 - 


0.02195770 

0.01428322 


Potassium Bromide 

0.007676 0.0156569 

n 2 = 2.361323 - 0.00311497X 2 - 0.000,000,058613X 4 +-—-+ 


X 2 


X 2 - 0.0324 


Potassium Chloride (for the ultraviolet and visible) 
n 2 = a 2 + , M \- . + , k\ 2 - h\ 4 , n 2 = b 2 4- M ‘ 


X 2 -X, 2 X 2 -X 2 2 


m 2 m 3 


X 2 - X, 2 X 2 - x 2 2 ' x 3 2 - 


a 

Mr 

X, 2 

M 2 


2 — 


2.174967 

0.008344206 

0.0119082 

0.00698382 


X 2 2 = 0.0255550 
k = 0.000513495 
h = 0.06167587 


b 2 = 3.866619 
M 3 = 5569.715 
X 3 2 = 3292.47 



















MATERIAL DATA USEFUL FOR LENS DESIGN 


337 


Rutile 

Ordinary ray: 

n 2 = 5.913 + 2.441 X 10 7 /(A 2 - 0.803 X 10 7 ) 
Extraordinary ray: 

n 2 = 7.197 + 3.322 X 10 7 /(A 2 - 0.843 X 10 7 ) 


Silver Chloride 

n 2 = 4.00804 - 0.0008511 IX 2 - 0.00000019762A 4 + 0.079086/( X 2 - 0.04584) 
Sphalerite 


n 2 = 5.164 + 1.208 X 10 7 /(A 2 - 0.732 X 10 7 ) 
Thallium Bromide-Iodide 


n 2 


i = 2 

i 


Ki \ 2 
X 2 - Xi 2 


i 

X, 2 

Ki 

1 

0.0225 

1.8293958 

2 

0.0625 

1.6675593 

3 

0.1225 

1.1210424 

4 

0.2025 

0.04513366 

5 

27089.737 

12.380234 


8.4.2. Herzberger Dispersion Equation. Herzberger and Salzberg [4] have given 
an equation which uses as X,, 2 the value 0.028 p 2 . The equation is 

n — A. + BL + CL 2 + Dk 2 + Ek 4 (8-1) 

L — (X 2 — 0.028) -1 . Values for the constants A to E and the range of usefulness of the 
equation are given in Table 8-14. Residuals between these data and measured values 
are ±3 X 10 -4 or less with a very few exceptions. 


Table 8-14. Constants to be Used with the Interpolation Formula 

Wavelength Constant 


No. 

Material 

Range (g) 








from 

to 

A 

B 

C 

D 

E 

1 

Fused quartz 

0.5 

4.3 

1.44902 

0.004604 

-0.000381 

-0.0025268 

-0.00007722 

2 

Calcium aluminate 

0.6 

4.3 

1.64289 

0.007860 

-0.000231 

-0.0022133 

-0.00001598 

3 

IR-20 

0.5 

5.0 

1.83450 

0.011834 

-0.000100 

-0.0022268 

-0.00001267 

4 

Strontium titanate 

1.0 

5.3 

2.28355 

0.035906 

+0.001666 

-0.0061335 

-0.00001502 

5 

Magnesium oxide 

0.5 

5.5 

1.71960 

0.006305 

-0.000090 

-0.0031356 

-0.00000770 

6 

Sapphire 

1.0 

5.6 

1.75458 

0.007149 

-0.001577 

-0.0045380 

-0.00002808 

7 

Lithium fluoride 

0.5 

6.0 

1.38761 

0.001796 

-0.000041 

-0.0023045 

-0.00000557 

8 

Irtran 1 

1.0 

6.7 

1.37770 

0.001348 

+0.000216 

-0.0015041 

-0.00000441 

9 

Calcium fluoride 

0.6 

8.3 

1.42780 

0.002267 

-0.000069 

-0.0011157 

-0.00000162 

10 

Barium fluoride 

0.5 

11.0 

1.46629 

0.002867 

+0.000064 

-0.0006035 

-0.000000465 

11 

Silicon 

1.3 

11.0 

3.41696 

0.138497 

+0.013924 

-0.0000209 

+0.000000148 

12 

Arsenic trisulfide 

0.6 

12.0 

2.41326 

0.055720 

+0.006177 

-0.0003044 

-0.000000232 

13 

Irtran 2 

1.0 

13.5 

2.25698 

0.032586 

+0.000679 

-0.0005272 

-0.000000604 

14 

Germanium 

2.0 

13.5 

3.99931 

0.391707 

+0.163492 

-0.0000060 

+0.000000053 





-VALUES 


338 


OPTICAL MATERIALS 


8.4.3. Refractive Index, Dispersion Values. Figure 8-2 shows the usual "glass 
table.” From such a plot the optical designer can choose glasses to balance powers 
and chromatic aberrations. For infrared lens design, a different Abbe number (abscissa 
of Fig. 8-2) must be defined. Three definitions appear to be useful: 

wi.ss - 1 

^ 1 - 2.7 — 

fl 2.7 — fl 1.0 


^ 4.25 1 

^ 3 - 5.5 — ~ 

n h . 5 — 1 


ttl 1.5 1 

^ 8-15 — 

n 15 — n s 


( 8 - 2 ) 


1000 


100 




10 


1.0 































< 

1 



















CsB 

r .CsI 







1 

CBr 



o r 

Si 

■ 






AgC 

| 

KR 

rk.n 

S-6 

J 

o — 0 

>e(As) 

—AS2S3 













■T Ge 



.. 1 



. J 






tsar 2 i 




I 








i 

<» 


_ 





CaF2 

k 








NaFf f 

I - MerO 








... . 


t 


k 







I 

u 

< 

, k 



• 

InSb 



Si 0 2 j ' 

Li 

> 

to 

o 

3 

SrT 

Ti 0 2 
i0 3 t 






__< 


it 






_ L _ XT 

nPl / 

"S T 






/ N 

dLI i 

IT 









i 





LiF 









































1.0 2.0 3.0 4.0 

REFRACTIVE INDEX 


A = y ^ m = v m = p 

1.0-2.7 3.0-5.5 8-15 

= visible values for optical glasses. 


Fig. 8-16. Reciprocal dispersion of some infrared optical materials. 














































































MATERIAL DATA USEFUL FOR LENS DESIGN 339 

Values are plotted in Fig. 8-16. The lines connect values of v in different spectral 
regions for the same material. 

Values of partial dispersion are also useful. They can be easily calculated from 
index data. A curve for 15 materials useful from 8 fx to 15 /x is given in Fig. 8-17. 
Such a curve, which is easily constructed for other materials, is useful in the design 
of highly color-corrected triplet lenses (P is the partial dispersion). Equations for 
the combined power of a system of thin lenses (Chapter 9) show that the differences in 
v of the first and last elements should be large, and that the change in P AP should 
be large. In this case AP is the difference between the line connecting the curves for 
two materials and the third material. 

Another useful technique invented by Szeles and Cuny [51 employs the concept of 
angular dispersion. Quantities are defined in Fig. 8-18, which also gives data for a 
number of useful materials. 



Fig. 8-17. Dispersive power of 15 materials for use in the infrared. 



ANGULAR DISPERSION 


340 


OPTICAL MATERIALS 



2 4 6 8 10 12 14 16 


WAVELENGTH (p) 

Fig. 8-18. The angular dispersion of a thin lens as a function of wavelength. 



























MATERIAL DATA USEFUL FOR LENS DESIGN 341 

8.4.4. G Sums. In lens design, Conrady [6] defines and uses terms called G sums. 
Values for the sums are listed in Table 8-15. 


G 0 =— (n — 1) 

G! = n 2 G 0 
G 2 = (2 n + 1 )G 
G 3 = (3 n + 1)G 




G 8 — nG o 



n 


Table 8-15. Conrady G Sums 


N 

Gi 

g 2 

G , 

g 4 

g 5 

G ti 

g 7 

G » 

g 9 

1.3500 

0.3189 

0.6475 

0.8837 

0.4343 

1.2185 

0.7843 

0.4796 

0.2362 

0.1296 

.3600 

.3329 

.6696 

.9144 

.4447 

.2494 

.8047 

.4924 

.2448 

.1324 

.3700 

.3472 

.6919 

.9453 

.4551 

.2801 

.8251 

.5050 

.2534 

.1350 

.3800 

.3618 

.7144 

.9766 

.4654 

.3107 

.8454 

.5177 

.2622 

.1377 

.3900 

.3768 

.7371 

1.0081 

.4756 

.3412 

.8656 

.5303 

.2710 

.1403 

.4000 

.3920 

.7600 

.0400 

.4857 

.3714 

.8857 

.5429 

.2800 

.1429 

.4100 

.4076 

.7831 

.0721 

.4958 

.4016 

.9058 

.5554 

.2890 

.1454 

.4200 

.4234 

.8064 

.1046 

.5058 

.4315 

.9258 

.5679 

.2982 

.1479 

.4300 

.4397 

.8299 

.1373 

.5157 

.4614 

.9457 

.5803 

.3074 

.1503 

.4400 

.4562 

.8536 

.1704 

.5256 

.4911 

.9656 

.5928 

.3168 

.1528 

.4500 

.4731 

.8775 

.2037 

.5353 

.5207 

.9853 

.6052 

.3262 

.1552 

.4600 

.4903 

.9016 

.2374 

.5451 

.5501 

1.0051 

.6175 

.3358 

.1575 

.4700 

.5078 

.9259 

.2713 

.5547 

.5795 

.0247 

.6299 

.3454 

.1599 

.4800 

.5257 

.9504 

.3056 

.5643 

.6086 

.0443 

.6422 

.3552 

.1622 

.4900 

.5439 

.9751 

.3401 

.5739 

.6377 

.0639 

.6544 

.3650 

.1644 

.5000 

.5625 

1.0000 

.3750 

.5833 

.6667 

.0833 

.6667 

.3750 

.1667 

.5100 

.5814 

.0251 

.4101 

.5927 

.6955 

.1027 

.6789 

.3850 

.1689 

.5200 

.6007 

.0504 

.4456 

.6021 

.7242 

.1221 

.6911 

.3952 

.1711 

.5300 

.6203 

.0759 

.4813 

.6114 

.7528 

.1414 

.7032 

.4054 

.1732 

.5400 

.6403 

.1016 

.5174 

.6207 

.7813 

.1606 

.7153 

.4158 

.1753 

.5500 

.6607 

.1275 

.5537 

.6298 

.8097 

.1798 

.7274 

.4262 

.1774 

.5600 

.6814 

.1536 

.5904 

.6390 

.8379 

.1990 

.7395 

.4368 

.1795 

.5700 

.7025 

.1799 

.6273 

.6481 

.8661 

.2181 

.7515 

.4474 

.1815 

.5800 

.7240 

.2064 

.6646 

.6571 

.8942 

.2371 

.7635 

.4582 

.1835 

.5900 

.7458 

.2331 

.7021 

.6661 

.9221 

.2561 

.7755 

.4690 

.1855 

.6000 

.7680 

.2600 

.7400 

.6750 

.9500 

.2750 

.7875 

.4800 

.1875 

.6100 

.7906 

.2871 

.7781 

.6839 

.9778 

.2939 

.7994 

.4910 

.1894 

.6200 

.8136 

.3144 

.8166 

.6927 

2.0054 

.3127 

.8114 

.5022 

.1914 

.6300 

.8369 

.3419 

.8553 

.7015 

.0330 

.3315 

.8233 

.5134 

.1933 

.6400 

.8607 

.3696 

.8944 

.7102 

.0605 

.3502 

.8351 

.5248 

.1951 

.6500 

.8848 

.3975 

.9337 

.7189 

.0879 

.3689 

.8470 

.5362 

.1970 

.6600 

.9093 

.4256 

.9734 

.7276 

.1152 

.3876 

.8588 

.5478 

.1988 

.6700 

.9343 

.4539 

2.0133 

.7362 

.1424 

.4062 

.8706 

.5594 

.2006 

.6800 

.9596 

.4824 

.0536 

.7448 

.1695 

.4248 

.8824 

.5712 

.2024 

.6900 

.9854 

.5111 

.0941 

.7533 

.1966 

.4433 

.8941 

.5830 

.2041 






342 


OPTICAL MATERIALS 


Table 8-15. Conrady G Sums ( Continued ) 


N 

G i 

g 2 

G3 

g 4 

G $ 

Ge 

g 7 

G 8 

g 9 

1.7000 

1.0115 

1.5400 

2.1350 

0.7618 

2.2235 

1.4618 

0.9059 

0.5950 

0.2059 

.7100 

.0381 

.5691 

.1761 

.7702 

.2504 

.4802 

.9176 

.6070 

.2076 

.7200 

.0650 

.5984 

.2176 

.7786 

.2772 

.4986 

.9293 

.6192 

.2093 

.7300 

.0924 

.6279 

.2593 

.7870 

.3039 

.5170 

.9410 

.6314 

.2110 

.7400 

.1202 

.6576 

.3014 

.7953 

.3306 

.5353 

.9526 

.6438 

.2126 

.7500 

.1484 

.6875 

.3437 

.8036 

.3571 

.5536 

.9643 

.6562 

.2143 

.7600 

.1771 

.7176 

.3864 

.8118 

.3836 

.5718 

.9759 

.6688 

.2159 

.7700 

.2062 

.7479 

.4293 

.8200 

.4101 

.5900 

.9875 

.6814 

.2175 

.7800 

.2357 

.7784 

.4726 

.8282 

.4364 

.6082 

.9991 

.6942 

.2191 

.7900 

.2656 

.8091 

.5161 

.8363 

.4627 

.6263 

1.0107 

.7070 

.2207 

.8000 

.2960 

.8400 

.5600 

.8444 

.4889 

.6444 

.0222 

.7200 

.2222 

.8100 

.3268 

.8711 

.6041 

.8525 

.5150 

.6625 

.0338 

.7330 

.2238 

.8200 

.3581 

.9024 

.6486 

.8605 

.5411 

.6805 

.0453 

.7462 

.2253 

.8300 

.3898 

.9339 

.6933 

.8686 

.5671 

.6986 

.0568 

.7594 

.2268 

.8400 

.4220 

.9656 

.7384 

.8765 

.5930 

.7165 

.0683 

.7728 

.2283 

.8500 

.4546 

.9975 

.7837 

.8845 

.6189 

.7345 

.0797 

.7862 

.2297 

.8600 

.4876 

2.0296 

.8294 

.8924 

.6447 

.7524 

.0912 

.7998 

.2312 

.8700 

.5212 

.0619 

.8753 

.9002 

.6705 

.7702 

.1026 

.8134 

.2326 

.8800 

.5551 

.0944 

.9216 

.9081 

.6962 

.7881 

.1140 

.8272 

.2340 

.8900 

.5896 

.1271 

.9681 

.9159 

.7218 

.8059 

.1254 

.8410 

.2354 

.9000 

.6245 

.1600 

3.0150 

.9237 

.7474 

.8237 

.1368 

.8550 

.2368 

.9100 

.6599 

.1931 

.0621 

.9314 

.7729 

.8414 

.1482 

.8690 

.2382 

.9200 

.6957 

.2264 

.1096 

.9392 

.7983 

.8592 

.1596 

.8832 

.2396 

.9300 

.7321 

.2599 

.1573 

.9469 

.8237 

.8769 

.1709 

.8974 

.2409 

.9400 

.7689 

.2936 

.2054 

.9545 

.8491 

.8945 

.1823 

.9118 

.2423 

.9500 

.8062 

.3275 

.2537 

.9622 

.8744 

.9122 

.1936 

.9262 

.2436 

.9600 

.8440 

.3616 

.3024 

.9698 

.8996 

.9298 

.2049 

.9408 

.2449 

.9700 

.8822 

.3959 

.3513 

.9774 

.9248 

.9474 

.2162 

.9554 

.2462 

.9800 

.9210 

.4304 

.4006 

.9849 

.9499 

.9649 

.2275 

.9702 

.2475 

.9900 

.9602 

.4651 

.4501 

.9925 

.9750 

.9825 

.2387 

.9850 

.2487 

2.0000 

2.0000 

.5000 

.5000 

1.0000 

3.0000 

2.0000 

.2500 

1.0000 

.2500 

.0100 

.0403 

.5351 

.5501 

.0075 

.0250 

.0175 

.2612 

.0150 

.2512 

.0200 

.0810 

.5704 

.6006 

.0150 

.0499 

.0350 

.2725 

.0302 

.2525 

.0300 

.1223 

.6059 

.6513 

.0224 

.0748 

.0524 

.2837 

.0454 

.2537 

.0400 

.1640 

.6416 

.7024 

.0298 

.0996 

.0698 

.2949 

.0608 

.2549 

.0500 

.2063 

.6775 

.7537 

.0372 

.1244 

.0872 

.3061 

.0762 

.2561 

.0600 

.2491 

.7136 

.8054 

.0446 

.1491 

.1046 

.3173 

.0918 

.2573 

.0700 

.2924 

.7499 

.8573 

.0519 

.1738 

.1219 

.3285 

.1074 

.2585 

.0800 

.3363 

.7864 

.9096 

.0592 

.1985 

.1392 

.3396 

.1232 

.2596 

.0900 

.3806 

.8231 

.9621 

.0665 

.2231 

.1565 

.3508 

.1390 

.2608 

.1000 

.4255 

.8600 

4.0150 

.0738 

.2476 

.1738 

.3619 

.1550 

.2619 

.1100 

.4709 

.8971 

.0681 

.0811 

.2721 

.1911 

.3730 

.1710 

.2630 

.1200 

.5169 

.9344 

.1216 

.0883 

.2966 

.2083 

.3842 

.1872 

.2642 

.1300 

.5633 

.9719 

.1753 

.0955 

.3210 

.2255 

.3953 

.2034 

.2653 

.1400 

.6104 

3.0096 

.2294 

.1027 

.3454 

.2427 

.4064 

.2198 

.2664 

.1500 

.6579 

.0475 

.2837 

.1099 

.3698 

.2599 

.4174 

.2362 

.2674 

.1600 

.7060 

.0856 

.3384 

.1170 

.3941 

.2770 

.4285 

.2528 

.2685 

.1700 

.7547 

.1239 

.3933 

.1242 

.4183 

.2942 

.4396 

.2694 

.2696 

.1800 

.8039 

.1624 

.4486 

.1313 

.4426 

.3113 

.4506 

.2862 

.2706 

.1900 

.8537 

.2011 

.5041 

.1384 

.4668 

.3284 

.4617 

.3030 

.2717 


MATERIAL DATA USEFUL FOR LENS DESIGN 


343 


Table 8-15. Conrady G Sums ( Continued) 


N 

G , 

G 2 

G , 

G .4 

g 5 

G 6 

G ^ 

g 8 

Gy 

2.2000 

2.9040 

3.2400 

4.5600 

1.1455 

3.4909 

2.3455 

1.4727 

1.3200 

0.2727 

.2100 

.9549 

.2791 

.6161 

.1525 

.5150 

.3625 

.4838 

.3370 

.2738 

.2200 

3.0063 

.3184 

.6726 

.1595 

.5391 

.3795 

.4948 

.3542 

.2748 

.2300 

.0583 

.3579 

.7293 

.1666 

.5631 

.3966 

.5058 

.3714 

.2758 

.2400 

.1109 

.3976 

.7864 

.1736 

.5871 

.4136 

.5168 

.3888 

.2768 

.2500 

.1641 

.4375 

.8437 

.1806 

.6111 

.4306 

.5278 

.4062 

.2778 

.2600 

.2178 

.4776 

.9014 

.1875 

.6350 

.4475 

.5388 

.4238 

.2788 

.2700 

.2721 

.5179 

.9593 

.1945 

.6589 

.4645 

.5497 

.4414 

.2797 

.2800 

.3270 

.5584 

5.0176 

.2014 

.6828 

.4814 

.5607 

.4592 

.2807 

.2900 

.3824 

.5991 

.0761 

.2083 

.7066 

.4983 

.5717 

.4770 

.2817 

.3000 

.4385 

.6400 

.1350 

.2152 

.7304 

.5152 

.5826 

.4950 

.2826 

.3100 

.4951 

.6811 

.1941 

.2221 

.7542 

.5321 

.5935 

.5130 

.2835 

.3200 

.5524 

.7224 

.2536 

.2290 

.7779 

.5490 

.6045 

.5312 

.2845 

.3300 

.6102 

.7639 

.3133 

.2358 

.8016 

.5658 

.6145 

.5494 

.2854 

.3400 

.6687 

.8056 

.3734 

.2426 

.8253 

.5826 

.6263 

.5678 

.2863 

.3500 

.7277 

.8475 

.4337 

.2495 

.8489 

.5995 

.6372 

.5862 

.2872 

.3600 

.7873 

.8896 

.4944 

.2563 

.8725 

.6163 

.6481 

.6048 

.2881 

.3700 

.8476 

.9319 

.5553 

.2631 

.8961 

.6331 

.6590 

.6234 

.2890 

.3800 

.9084 

.9744 

.6166 

.2698 

.9197 

.6498 

.6699 

.6422 

.2899 

.3900 

.9699 

4.0171 

.6781 

.2766 

.9432 

.6666 

.6808 

.6610 

.2908 

.4000 

4.0320 

.0600 

.7400 

.2833 

.9667 

.6833 

.6917 

.6800 

.2917 

.4100 

.0947 

.1031 

.8021 

.2901 

.9901 

.7001 

.7025 

.6990 

.2925 

.4200 

.1580 

.1464 

.8646 

.2968 

4.0136 

.7168 

.7134 

.7182 

.2934 

.4300 

.2220 

.1899 

.9273 

.3035 

.0370 

.7335 

.7242 

.7374 

.2942 

.4400 

.2866 

.2336 

.9904 

.3102 

.0603 

.7502 

.7351 

.7568 

.2951 

.4500 

.3518 

.2775 

6.0537 

.3168 

.0837 

.7668 

.7459 

.7762 

.2959 

.4600 

.4177 

.3216 

.1174 

.3235 

.1070 

.7835 

.7567 

.7958 

.2967 

.4700 

.4842 

.3659 

.1813 

.3301 

.1303 

.8001 

.7676 

.8154 

.2976 

.4800 

.5513 

.4104 

.2456 

.3368 

.1535 

.8168 

.7784 

.8352 

.2984 

.4900 

.6191 

.4551 

.3101 

.3434 

.1768 

.8334 

.7892 

.8550 

.2992 

.5000 

.6875 

.5000 

.3750 

.3500 

.2000 

.8500 

.8000 

.8750 

.3000 

.5100 

.7566 

.5451 

.4401 

.3566 

.2232 

.8666 

.8108 

.8950 

.3008 

.5200 

.8263 

.5904 

.5056 

.3632 

.2463 

.8832 

.8216 

.9152 

.3016 

.5300 

.8967 

.6359 

.5713 

.3697 

.2695 

.8997 

.8324 

.9354 

.3024 

.5400 

.9677 

.6816 

.6374 

.3763 

.2926 

.9163 

.8431 

.9558 

.3031 

.5500 

5.0394 

.7275 

.7037 

.3828 

.3157 

.9328 

.8539 

.9762 

.3039 

.5600 

.1118 

.7736 

.7704 

.3894 

.3387 

.9494 

.8647 

.9968 

.3047 

.5700 

.1848 

.8199 

.8373 

.3959 

.3618 

.9659 

.8754 

2.0174 

.3054 

.5800 

.2586 

.8664 

.9046 

.4024 

.3848 

.9824 

.8862 

.0382 

.3062 

.5900 

.3329 

.9131 

.9721 

.4089 

.4078 

.9989 

.8969 

.0590 

.3069 

.6000 

.4080 

.9600 

7.0400 

.4154 

.4308 

3.0154 

.9077 

.0800 

.3077 

.6100 

.4837 

5.0071 

.1081 

.4219 

.4537 

.0319 

.9184 

.1010 

.3084 

.6200 

.5602 

.0544 

.1766 

.4283 

.4766 

.0483 

.9292 

.1222 

.3092 

.6300 

.6373 

.1019 

.2453 

.4348 

.4995 

.0648 

.9399 

.1434 

.3099 

.6400 

.7151 

.1496 

.3144 

.4412 

.5224 

.0812 

.9506 

.1648 

.3106 

.6500 

.7936 

.1975 

.3837 

.4476 

.5453 

.0976 

.9613 

.1862 

.3113 

.6600 

.8727 

.2456 

.4534 

.4541 

.5681 

.1141 

.9720 

.2078 

.3120 

.6700 

.9526 

.2939 

.5233 

.4605 

.5909 

.1305 

.9827 

.2294 

.3127 

.6800 

6.0332 

.3424 

.5936 

.4669 

.6137 

.1469 

.9934 

.2512 

.3134 

.6900 

.1145 

.3911 

.6641 

.4733 

.6365 

.1633 

2.0041 

.2730 

.3141 


344 


OPTICAL MATERIALS 


Table 8-15. Conrady G Sums ( Continued ) 


N 

G , 

g 2 

g 3 

g 4 

g 5 

g 6 

g 7 

Gs 

g 9 

2.7000 

6.1965 

5.4400 

7.7350 

1.4796 

4.6593 

3.1796 

2.0148 

2.2950 

0.3148 

.7100 

.2792 

.4891 

.8061 

.4860 

.6820 

.1960 

.0255 

.3170 

.3155 

.7200 

.3626 

.5384 

.8776 

.4924 

.7047 

.2124 

.0362 

.3392 

.3162 

.7300 

.4468 

.5879 

.9493 

.4987 

.7274 

.2287 

.0468 

.3614 

.3168 

.7400 

.5316 

.6376 

8.0214 

.5050 

.7501 

.2450 

.0575 

.3838 

.3175 

.7500 

.6172 

.6875 

.0937 

.5114 

.7727 

.2614 

.0682 

.4062 

.3182 

.7600 

.7035 

.7376 

.1664 

.5177 

.7954 

.2777 

.0788 

.4288 

.3188 

.7700 

.7905 

.7879 

.2393 

.5240 

.8180 

.2940 

.0895 

.4514 

.3195 

.7800 

.8783 

.8384 

.3126 

.5303 

.8406 

.3103 

.1001 

.4742 

.3201 

.7900 

.9668 

.8891 

.3861 

.5366 

.8632 

.3266 

.1108 

.4970 

.3208 

.8000 

7.0560 

.9400 

.4600 

.5429 

.8857 

.3429 

.1214 

.5200 

.3214 

.8100 

.1460 

.9911 

.5341 

.5491 

.9083 

.3591 

.1321 

.5430 

.3221 

.8200 

.2367 

6.0424 

.6086 

.5554 

.9308 

.3754 

.1427 

.5662 

.3227 

.8300 

.3281 

.0939 

.6833 

.5616 

.9533 

.3916 

.1533 

.5894 

.3233 

.8400 

.4204 

.1456 

.7584 

.5679 

.9758 

.4079 

.1639 

.6128 

.3239 

.8500 

.5133 

.1975 

.8337 

.5741 

.9982 

.4241 

.1746 

.6362 

.3246 

.8600 

.6070 

.2496 

.9094 

.5803 

5.0207 

.4403 

.1852 

.6598 

.3252 

.8700 

.7015 

.3019 

.9853 

.5866 

.0431 

.4566 

.1958 

.6834 

.3258 

.8800 

.7967 

.3544 

9.0616 

.5928 

.0656 

.4728 

.2064 

.7072 

.3264 

.8900 

.8927 

.4071 

.1381 

.5990 

.0880 

.4890 

.2170 

.7310 

.3270 

.9000 

.9895 

.4600 

.2150 

.6052 

.1103 

.5052 

.2276 

.7550 

.3276 

.9100 

8.0870 

.5131 

.2921 

.6114 

.1327 

.5214 

.2382 

.7790 

.3282 

.9200 

.1853 

.5664 

.3696 

.6175 

.1551 

.5375 

.2488 

.8032 

.3288 

.9300 

.2844 

.6199 

.4473 

.6237 

.1774 

.5537 

.2594 

.8274 

.3294 

.9400 

.3843 

.6736 

.5254 

.6299 

.1997 

.5699 

.2699 

.8518 

.3299 

.9500 

.4849 

.7275 

.6037 

.6360 

.2220 

.5860 

.2805 

.8762 

.3305 

.9600 

.5864 

.7816 

.6824 

.6422 

.2443 

.6022 

.2911 

.9008 

.3311 

.9700 

.6886 

.8359 

.7613 

.6483 

.2666 

.6183 

.3016 

.9254 

.3316 

.9800 

.7916 

.8904 

.8406 

.6544 

.2889 

.6344 

.3122 

.9502 

.3322 

.9900 

.8954 

.9451 

.9201 

.6606 

.3111 

.6506 

.3228 

.9750 

.3328 

3.0000 

9.0000 

7.0000 

10.0000 

.6667 

.3333 

.6667 

.3333 

3.0000 

.3333 

.0100 

.1054 

.0551 

.0801 

.6728 

.3555 

.6828 

.3439 

.0250 

.3339 

.0200 

.2116 

.1104 

.1606 

.6789 

.3777 

.6989 

.3544 

.0502 

.3344 

.0300 

.3186 

.1659 

.2413 

.6850 

.3999 

.7150 

.3650 

.0754 

.3350 

.0400 

.4264 

.2216 

.3224 

.6911 

.4221 

.7311 

.3755 

.1008 

.3355 

.0500 

.5351 

.2775 

.4037 

.6971 

.4443 

.7471 

.3861 

.1262 

.3361 

.0600 

.6445 

.3336 

.4854 

.7032 

.4664 

.7632 

.3966 

.1518 

.3366 

.0700 

.7548 

.3899 

.5673 

.7093 

.4885 

.7793 

.4071 

.1774 

.3371 

.0800 

.8659 

.4464 

.6496 

.7153 

.5106 

.7953 

.4177 

.2032 

.3377 

.0900 

.9778 

.5031 

.7321 

.7214 

.5328 

.8114 

.4282 

.2290 

.3382 

.1000 

10.0905 

.5600 

.8150 

.7274 

.5548 

.8274 

.4387 

.2550 

.3387 

.1100 

.2041 

.6171 

.8981 

.7335 

.5769 

.8435 

.4492 

.2810 

.3392 

.1200 

.3185 

.6744 

.9816 

.7395 

.5990 

.8595 

.4597 

.3072 

.3397 

.1300 

.4337 

.7319 

11.0653 

.7455 

.6210 

.8755 

.4703 

.3334 

.3403 

.1400 

.5498 

.7896 

.1494 

.7515 

.6431 

.8915 

.4808 

.3598 

.3408 

.1500 

.6667 

.8475 

.2337 

.7575 

.6651 

.9075 

.4913 

.3862 

.3413 

.1600 

.7844 

.9056 

.3184 

.7635 

.6871 

.9235 

.5018 

.4128 

.3418 

.1700 

.9031 

.9639 

.4033 

.7695 

.7091 

.9395 

.5123 

.4394 

.3423 

.1800 

11.0225 

8.0224 

.4886 

.7755 

.7311 

.9555 

.5228 

.4662 

.3428 

.1900 

.1428 

.0811 

.5741 

.7815 

.7530 

.9715 

.5333 

.4930 

.3433 


MATERIAL DATA USEFUL FOR LENS DESIGN 


345 


Table 8-15. Conrady G Sums ( Continued ) 


N 

G , 

g 2 

g 3 

g 4 

G$ 

Ge 

g 7 

Gg 

Gg 

3.2000 

11.2640 

8.1400 

11.6600 

1.7875 

5.7750 

3.9875 

2.5437 

3.5200 

0.3437 

.2100 

.3860 

.1991 

.7461 

.7935 

.7969 

4.0035 

.5542 

.5470 

.3442 

.2200 

.5089 

.2584 

.8326 

.7994 

.8189 

.0194 

.5647 

.5742 

.3447 

.2300 

.6327 

.3179 

.9193 

.8054 

.8408 

.0354 

.5752 

.6014 

.3452 

.2400 

.7573 

.3776 

12.0064 

.8114 

.8627 

.0514 

.5857 

.6288 

.3457 

.2500 

.8828 

.4375 

.0937 

.8173 

.8846 

.0673 

.5962 

.6562 

.3462 

.2600 

12.0092 

.4976 

.1814 

.8233 

.9065 

.0833 

.6066 

.6838 

.3466 

.2700 

.1364 

.5579 

.2693 

.8292 

.9284 

.0992 

.6171 

.7114 

.3471 

.2800 

.2646 

.6184 

.3576 

.8351 

.9502 

.1151 

.6276 

.7392 

.3476 

.2900 

.3936 

.6791 

.4461 

.8410 

.9721 

.1310 

.6380 

.7670 

.3480 

.3000 

.5235 

.7400 

.5350 

.8470 

.9939 

.1470 

.6485 

.7950 

.3485 

.3100 

.6543 

.8011 

.6241 

.8529 

6.0158 

.1629 

.6589 

.8230 

.3489 

.3200 

.7860 

.8624 

.7136 

.8588 

.0376 

.1788 

.6694 

.8512 

.3494 

.3300 

.9186 

.9239 

.8033 

.8647 

.0594 

.1947 

.6798 

.8794 

.3498 

.3400 

.0521 

.9856 

.8934 

.8706 

.0812 

.2106 

.6903 

.9078 

.3503 

.3500 

13.1864 

9.0475 

.9837 

.8765 

.1030 

.2265 

.7007 

.9362 

.3507 

.3600 

.3217 

.1096 

13.0744 

.8824 

.1248 

.2424 

.7112 

.9648 

.3512 

.3700 

.4579 

.1719 

.1653 

.8883 

.1465 

.2583 

.7216 

.9934 

.3516 

.3800 

.5950 

.2344 

.2566 

.8941 

.1683 

.2741 

.7321 

4.0222 

.3521 

.3900 

.7331 

.2971 

.3481 

.9000 

.1900 

.2900 

.7425 

.0510 

.3525 

.4000 

.8720 

.3600 

.4400 

.9059 

.2118 

.3059 

.7529 

.0800 

.3529 

.4100 

14.0119 

.4231 

.5321 

.9117 

.2335 

.3217 

.7634 

.1090 

.3534 

.4200 

.1526 

.4864 

.6246 

.9176 

.2552 

.3376 

.7738 

.1382 

.3538 

.4300 

.2944 

.5499 

.7173 

.9235 

.2769 

.3535 

.7842 

.1674 

.3542 

.4400 

.4370 

.6136 

.8104 

.9293 

.2986 

.3693 

.7947 

.1968 

.3547 

.4500 

.5806 

.6775 

.9037 

.9351 

.3203 

.3851 

.8051 

.2262 

.3551 

.4600 

.7251 

.7416 

.9974 

.9410 

.3420 

.4010 

.8155 

.2558 

.3555 

.4700 

.8705 

.8059 

14.0913 

.9468 

.3636 

.4168 

.8259 

.2854 

.3559 

.4800 

15.0169 

.8704 

.1856 

.9526 

.3853 

.4326 

.8363 

.3152 

.3563 

.4900 

.1642 

.9351 

.2801 

.9585 

.4069 

.4485 

.8467 

.3450 

.3567 

.5000 

.3125 

10.0000 

.3750 

.9643 

.4286 

.4643 

.8571 

.3750 

.3571 

.5100 

.4617 

.0651 

.4701 

.9701 

.4502 

.4801 

.8675 

.4050 

.3575 

.5200 

.6119 

.1304 

.5656 

.9759 

.4718 

.4959 

.8780 

.4352 

.3580 

.5300 

.7630 

.1959 

.6613 

.9817 

.4934 

.5117 

.8884 

.4654 

.3584 

.5400 

.9151 

.2616 

.7574 

.9875 

.5150 

.5275 

.8988 

.4958 

.3588 

.5500 

16.0682 

.3275 

.8537 

.9933 

.5366 

.5433 

.9092 

.5262 

.3592 

.5600 

.2222 

.3936 

.9504 

.9991 

.5582 

.5591 

.9196 

.5568 

.3596 

.5700 

.3772 

.4599 

15.0473 

2.0049 

.5798 

.5749 

.9299 

.5874 

.3599 

.5800 

.5332 

.5264 

.1446 

.0107 

.6013 

.5907 

.9403 

.6182 

.3603 

.5900 

.6901 

.5931 

.2421 

.0164 

.6229 

.6064 

.9507 

.6490 

.3607 

.6000 

.8480 

.6600 

.3400 

.0222 

.6444 

.6222 

.9611 

.6800 

.3611 

.6100 

17.0069 

.7271 

.4381 

.0280 

.6660 

.6380 

.9715 

.7110 

.3615 

.6200 

.1668 

.7944 

.5366 

.0338 

.6875 

.6538 

.9819 

.7422 

.3619 

.6300 

.3276 

.8619 

.6353 

.0395 

.7090 

.6695 

.9923 

.7734 

.3623 

.6400 

.4895 

.9296 

.7344 

.0453 

.7305 

.6853 

3.0036 

.8048 

.3626 

.6500 

.6523 

.9975 

.8337 

.0510 

.7521 

.7010 

.0130 

.8362 

.3630 

.6600 

.8161 

11.0656 

.9334 

.0568 

.7736 

.7168 

.0234 

.8678 

.3634 

.6700 

.9810 

.1339 

16.0333 

.0625 

.7950 

.7325 

.0338 

.8994 

.3638 

.6800 

18.1468 

.2024 

.1336 

.0683 

.8165 

.7483 

.0441 

.9312 

.3641 

.6900 

.3137 

.2711 

.2341 

.0740 

.8380 

.7640 

.0545 

.9630 

.3645 


346 


OPTICAL MATERIALS 


Table 8-15. Conrady G Sums ( Continued ) 


N 

G , 

g 2 

g 3 

G 4 

g 5 

G 6 

g 7 

Gs 

g 9 

3.7000 

18.4815 

11.3400 

16.3350 

2.0797 

6.8595 

4.7797 

3.0649 

4.9950 

0.3649 

.7100 

.6504 

.4091 

.4361 

.0855 

.8809 

.7955 

.0752 

5.0270 

.3652 

.7200 

.8202 

.4784 

.5376 

.0912 

.9024 

.8112 

.0856 

.0592 

.3656 

.7300 

.9911 

.5479 

.6393 

.0969 

.9238 

.8269 

.0960 

.0914 

.3660 

.7400 

19.1630 

.6176 

.7414 

.1026 

.9452 

.8426 

.1063 

.1238 

.3663 

.7500 

.3359 

.6875 

.8437 

.1083 

.9667 

.8583 

.1167 

.1562 

.3667 

.7600 

.5099 

.7576 

.9464 

.1140 

.9881 

.8740 

.1270 

.1888 

.3670 

.7700 

.6849 

.8279 

17.0493 

.1197 

7.0095 

.8897 

.1374 

.2214 

.3674 

.7800 

.8609 

.8984 

.1526 

.1254 

.0309 

.9054 

.1477 

.2542 

.3677 

.7900 

20.0379 

.9691 

.2561 

.1311 

.0523 

.9211 

.1581 

.2870 

.3681 

.8000 

.2160 

12.0400 

.3600 

.1368 

.0737 

.9368 

.1684 

.3200 

.3684 

.8100 

.3951 

.1111 

.4641 

.1425 

.0951 

.9525 

.1788 

.3530 

.3688 

.8200 

.5753 

.1824 

.5686 

.1482 

.1164 

.9682 

.1891 

.3862 

.3691 

.8300 

.7565 

.2539 

.6733 

.1539 

.1378 

.9839 

.1995 

.4194 

.3695 

.8400 

.9388 

.3256 

.7784 

.1596 

.1592 

.9996 

.2098 

.4528 

.3698 

.8500 

21.1221 

.3975 

.8837 

.1653 

.1805 

5.0153 

.2201 

.4862 

.3701 

.8600 

.3064 

.4696 

.9894 

.1709 

.2019 

.0309 

.2305 

.5198 

.3705 

.8700 

.4919 

.5419 

18.0953 

.1766 

.2232 

.0466 

.2408 

.5534 

.3708 

.8800 

.6783 

.6144 

.2016 

.1823 

.2445 

.0623 

.2511 

.5872 

.3711 

.8900 

.8659 

.6871 

.3081 

.1879 

.2659 

.0779 

.2615 

.6210 

.3715 

.9000 

22.0545 

.7600 

.4150 

.1936 

.2872 

.0936 

.2718 

.6550 

.3718 

.9100 

.2442 

.8331 

.5221 

.1992 

.3085 

.1092 

.2821 

.6890 

.3721 

.9200 

.4349 

.9064 

.6296 

.2049 

.3298 

.1249 

.2924 

.7232 

.3724 

.9300 

.6268 

.9799 

.7373 

.2105 

.3511 

.1405 

.3028 

.7574 

.3728 

.9400 

.8197 

13.0536 

.8454 

.2162 

.3724 

.1562 

.3131 

.7918 

.3731 

.9500 

23.0137 

.1275 

.9537 

.2218 

.3937 

.1718 

.3234 

.8262 

.3734 

.9600 

.2088 

.2016 

19.0624 

.2275 

.4149 

.1875 

.3337 

.8608 

.3737 

.9700 

.4049 

.2759 

.1713 

.2331 

.4362 

.2031 

.3441 

.8954 

.3741 

.9800 

.6022 

.3504 

.2806 

.2387 

.4575 

.2187 

.3544 

.9302 

.3744 

.9900 

.8005 

.4251 

.3901 

.2444 

.4787 

.2344 

.3647 

.9650 

.3747 

4.0000 

24.0000 

.5000 

.5000 

.2500 

.5000 

.2500 

.3750 

6.0000 

.3750 

.0100 

.2005 

.5751 

.6101 

.2556 

.5212 

.2656 

.3853 

.0350 

.3753 

.0200 

.4022 

.6504 

.7206 

.2612 

.5425 

.2812 

.3956 

.0702 

.3756 

.0300 

.6050 

.7259 

.8313 

.2669 

.5637 

.2969 

.4059 

.1054 

.3759 

.0400 

.8088 

.8016 

.9424 

.2725 

.5850 

.3125 

.4162 

.1408 

.3762 

.0500 

25.0138 

.8775 

20.0537 

.2781 

.6062 

.3281 

.4265 

.1762 

.3765 

.0600 

.2199 

.9536 

.1654 

.2837 

.6274 

.3437 

.4368 

.2118 

.3768 

.0700 

.4271 

14.0299 

.2773 

.2893 

.6486 

.3593 

.4471 

.2474 

.3771 

.0800 

.6355 

.1064 

.3896 

.2949 

.6698 

.3749 

.4575 

.2832 

.3775 

.0900 

.8449 

.1831 

.5021 

.3005 

.6910 

.3905 

.4678 

.3190 

.3778 

.1000 

26.0555 

.2600 

.6150 

.3061 

.7122 

.4061 

.4780 

.3550 

.3780 

.1100 

.2672 

.3371 

.7281 

.3117 

.7334 

.4217 

.4883 

.3910 

.3783 

.1200 

.4801 

.4144 

.8416 

.3173 

.7546 

.4373 

.4986 

.4272 

.3786 

.1300 

.6940 

.4919 

.9553 

.3229 

.7757 

.4529 

.5089 

.4634 

.3789 

.1400 

.9092 

.5696 

21.0694 

.3285 

.7969 

.4685 

.5192 

.4998 

.3792 

.1500 

27.1254 

.6475 

.1837 

.3340 

.8181 

.4840 

.5295 

.5362 

.3795 

.1600 

.3428 

.7256 

.2984 

.3396 

.8392 

.4996 

.5398 

.5728 

.3798 

.1700 

.5614 

.8039 

.4133 

.3452 

.8604 

.5152 

.5501 

.6094 

.3801 

.1800 

.7811 

.8824 

.5286 

.3508 

.8815 

.5308 

.5604 

.6462 

.3804 

.1900 

28.0020 

.9611 

.6441 

.3563 

.9027 

.5463 

.5707 

.6830 

.3807 

.2000 

.2240 

15.0400 

.7600 

.3619 

.9238 

.5619 

.5810 

.7200 

.3810 


USEFUL EQUATIONS 


347 


8.5. Useful Equations 

8.5.1. Reflection and Transmission at a Single Surface. S„ p + is the electric 
vector of a wave [7] traveling in the positive direction in the nth layer polarized with 
the electric vector parallel to the plane of incidence: S„,~ is the electric vector in the 
negative direction in the nth layer polarized perpendicular to the plane of incidence. 
Let n be n — ik. Then the Fresnel equations are: 


Eo P _ no cos </>i — ri\ cos </>o 
E 0p + n 0 cos <}>i + rii cos (f >o 

Ei p + _ 2n 0 cos </>o 

E 0p + n 0 cos </>i -f rii cos (f>o 


(8-3) 

(8-4) 


Eqs no cos </> 0 Tii cos <f>i 

S 0s + n 0 cos </>o + Th cos (f) t 


(8-5) 


Ei s + __ 2n 0 cos 4>o _ 

Eos + n 0 cos <f>o + rii cos <f>i 


( 8 - 6 ) 


Sometimes a more useful form for amplitude relations is 


tan (4>i — <t>o) 
tan (</>i + 4>o) 


(8-7) 


t in 


2 sin </>i cos 4> 0 


( 8 - 8 ) 


sin (<f >i — (f> o) 
sin (4 >i -I- </> 0 ) 


(8-9) 


2 sin </>i cos (f>o 
sin (</>i + 0 O ) 


For intensity, 


and 


(■ Eo P ~) 2 
(£o/) 2 


Pv = 


(£ oD 2 

(So /) 2 






and the transmittances are given by 


ni(Eip + ) 2 

n 0 (S 0/ /) 2 

Til (Si/) 2 
n 0 (So/) 2 


2 

11 p 

n 0 




> 


ni _ 

Is 

n 0 j 


— t- 2 


( 8 - 10 ) 


( 8 - 11 ) 


and 


( 8 - 12 ) 






















348 

Then for normal incidence, 


OPTICAL MATERIALS 


/n 0 — n A 2 

Pp — Ps — \ ) 

\n o 4 nJ 


n 0 -t- n i> 
4 n 0 /ii 


(8-13) 


Tp = T g 


(n 0 4- n i) 2 

For absorbing layers the equations are complicated. For normal incidence, 


n 0 — n i 4 iki 

n P = r u = -—-rr 

n 0 4 n\ — ik i 


which gives, for the reflectance of the surface 


When n 2 4 k 2 >> 1, 


Pp — p s — 


(no — ni) 2 4 ki 2 
(no 4 n x ) 2 4 ki 2 


(n 2 4 k 2 ) cos 2 </>o — 2n cos </>o 4 1 
(n 2 4 & 2 ) cos 2 </>o 4 2n cos <£ 0 4 1 


(n 2 4 /j 2 ) — 2n cos </>o 4 cos 2 <^>o 
(n 2 4 /s 2 ) 4 2n cos <£ 0 4 cos 2 </> 0 


(8-14) 


(8-15) 

(8-16) 

(8-17) 


8.5.2. Reflection and Transmission by a Single Layer. When multiple traversals 
are taken into account, the amplitude reflection coefficient is 


or 


r = r x 4 txt'ir 2 e~ 2i6 i — tit'irir 2 2 e~ 4i ^ 4 . . . 

tit' ir 2 e~ 2iS i 
r ' + 1 4 r x r 2 e~ 2i6 i 


(8-18) 


The amplitude transmission coefficient is 

t = tit 2 e ~' 6 > — tit 2 r t r 2 e ~ 3iS > 4 tit 2 r x 2 r 2 2 e ~ 5iS > — . . . 


tit 2 e ~ i6 « 

1 4 r x r 2 e~ 2ih i 


(8-19) 


To account for polarization differences, expressions for t x , t 2 , r it and r 2 should be ob¬ 
tained from the proper equations (8-3 to 8-6). The intensity ratios are 


T] 2 4 2r x r 2 cos 28i 4 r 2 2 
1 4 2r x r 2 cos 28i 4 ri 2 r 2 2 


( 8 - 20 ) 


_ n 2 ti 2 t 2 2 

no (1 4 2rir 2 cos 28i 4 ri 2 r 2 2 ) 

where 8i is (2 / nl\)n\di cos <f>i. For normal incidence, 

(n 0 2 4 n x 2 )(ni 2 4 n 2 2 ) — 4n 0 ni 2 n 2 4 (n 0 2 — n x 2 )(ni 2 — n 2 2 ) cos 28i 
(n 0 2 4 n\ 2 )(ni 2 4 n 2 2 ) 4 4non x 2 n 2 4 (n 0 2 — n x 2 )(ni 2 — n 2 2 ) cos 28 1 

8n 0 ni 2 n 2 

(no 2 4 n x 2 )(n\ 2 4 n 2 2 ) 4 4n 0 ni 2 n 2 4 (n 0 2 — n x 2 )(n x 2 — n 2 2 ) cos 28i 


( 8 - 21 ) 


( 8 - 22 ) 

(8-23) 


For absorbing media n must be replaced by n. 















USEFUL EQUATIONS 349 

8.5.3. Emissivity, Transmissivity, and Reflectivity for Partially Transparent 
Bodies [8]. The emissivity, e, apparent (measured) reflectance, p m , and apparent (mea¬ 
sured) transmittance, r m , of partially transparent bodies can be related as follows [81: 


€ + T m + p m = 1 (8-24) 

These terms are given below, where the functional dependence of all quantities 
on temperature and wavelength is suppressed. 


(1 -p)(l -t) 
1 — pr 


(8-25) 


pm 


{ PT 2 (1 ~ p) 2 
P 1 - p 2 r 2 


(8-26) 


Tm — T 


(1 ~P) 2 

1 — p 2 r 2 


(8-27) 


Figure 8-19 is a plot of Eq. (8-24). The complete thermal radiation properties of 
a given body at a particular wavelength and for a particular temperature are repre¬ 
sented by a single point on this chart. The location of the point can be determined 
experimentally when any two of the three quantities e, r m , and p m are given. From 
this, values of r and p can be obtained. 



Fig. 8-19. Thermal radiation chart for determining true values of transmissivity 
r and reflectivity p from experimentally measured values of emissivity e, apparent 
transmissivity r m , and apparent reflectivity p m . 























































350 


OPTICAL MATERIALS 



Fig. 8-20. Reflection loss for different 
incidence angles and different refrac¬ 
tive indices. 


Figure 8-20 illustrates Fresnel’s laws of reflection. It shows reflectivity p as a func¬ 
tion of angle for different refractive indices. The lower dashed curve shows p p , and the 
upper dashed curve p s , where subscripts p and s indicate the parallel and the perpen¬ 
dicular components of the electric field, respectively. The curve between them is 
characteristic of unpolarized light at the angles indicated for n = 1.5. The other curves 
are also for unpolarized light. The equations and curves show that there is a phase 
change of tt for E s vibrations of all angles of incidence. There is a phase change for 
E p vibrations only for angles equal to or exceeding the polarizing angle. For normal 
incident radiation the equation for reflectivity reduces to 

(n — 1\ 2 

P = (^) (8-28) 


Finally, at grazing incidence angles, the reflectivity approaches 100%. 
The transmittance for multiple reflections is 

(1 - p) 2 e~ ax 

T m 

\ __ p2 e -2ax 

If a is small, then 

(1-p) 2 1-p 

Toc 1-p 2 1 + p 

2 n 

n 2 - 1-1 


(8-29) 


(8-30) 


where r* = transmission for infinite number of reflections. Also, 

(n - l ) 2 

P « 2 i 1 
n 2 -f 1 

8.5.4. Loss Tangent. The loss tangent is defined as 


Ic 

Id €Oj 


„ -c cr 

tan 8 = — = — 


(8-31) 


(8-32) 


where /<• = conduction current 

la = displacement current 
a = conductivity 
€ = dielectric constant 
co = angular frequency 











OPTICAL SURFACE COATINGS 


351 

The loss tangent under proper assumptions is related to the absorption coefficient as 
follows: 

tan 8 = a\/27rn 

8.5.5. Extinction Coefficient. The extinction coefficient k is defined as 

k = akl4rr (8-33) 


8.6. Optical Surface Coatings 

8.6.1. Reflective Coatings. Aluminum, silver, gold, copper, rhodium, and titanium 
are the most frequently used mirror metals. Figure 8-21 shows the measured re¬ 
flectance of films of these metals. 



Fig. 8-21. Reflectance of various films of silver 
(Ag), gold (Au), aluminum (Al), copper (Cu), 
rhodium (Rh), and titanium (Ti) as a function 
of wavelength [9]. 

Stellite* cobalt-base alloy is often used when the surface must be exposed to the 
elements. It resists tarnish and corrosion and can be cleaned easily with a chalk- 
acetone paste. The reflectivity of Stellite* is given in Fig. 8-22 [10]. 

The most frequently used high-reflecting coating for first-surface mirrors is vacuum- 
deposited aluminum. It adheres better to glass and other substrates than the other 
high reflecting materials, does not tarnish in normal air, and is very easy to evaporate. 

The effect of the speed of evaporation or deposition rate on the reflectance of alumi¬ 
num coatings is considerable at visible and shorter wavelengths; however, in the 



* Registered trademark of Union Carbide Corporation. 







352 


OPTICAL MATERIALS 


infrared this becomes much less pronounced (Fig. 8-23). Almost any evaporation speed 
within reasonable limits results in aluminum coatings with more than 95% reflectance 
between 2 and 10 fx. It is always advisable, however, to evaporate aluminum as fast 
as possible because fast-deposited films are chemically and mechanically more durable. 



Fig. 8-23. Reflectance of two aluminum films 
prepared under extremely different evaporation 
conditions [9]. 


8.6.2. Filter Mirrors [9]. There are two film combinations that provide high 
infrared but low visible reflectance. These are: 

(1) Aluminum coated with evaporated germanium monoxide and silicon monoxide, 
each film approximately one quarter wavelength thick (Fig. 8-24). When germanium 
is evaporated onto opaque aluminum, the aluminum reflectance, controlled at a certain 
wavelength in the visible, decreases to a minimum of 30% to 40%. The addition of 
silicon monoxide decreases the visible reflectance to almost zero. Figure 8-25 shows 
the reflectance of two such mirrors. 




SiO to R =0% 





Al R thru 0 to 15% 




SiO to R 

min 



AI Opaque 




Fig. 8-24. Layer arrangement in (Al-SiO) 2 
filter mirror [9]. 


(2) Two silicon monoxide coatings, separated by a semitransparent aluminum film 
on top of the opaque aluminum mirror coating (Fig. 8-26). The first silicon monoxide 
film is evaporated until the aluminum reflectance reaches a minimum of about 60%. 
Then the aluminum is deposited until the reflectance, after passing through zero, rises 
again to a value of 15% to 20%. Another film of silicon monoxide on top brings the 
reflectance again down to zero. Figure 8-27 shows the reflectance of this type of 
reflector in the visible and infrared. 















OPTICAL SURFACE COATINGS 


353 



Fig. 8-25. Reflectance of Al-SiO filter mir¬ 
ror as a function of wavelength. Layer 
deposition controlled 0.55 /x [9]. 



Fig. 8-26. Layer arrangement in Al-SiO-Al-SiO filter 
mirror [9]. 



WAVELENGTH (/x) 


Fig. 8-27. Reflectance of Al-SiO-Al-SiO filter 
mirror as a function of wavelength. Layer 
deposition controlled at 550 m/x [9]. 












354 


OPTICAL MATERIALS 


8.6.3. Protective Coatings [9]. For many mirrors, the natural oxide film that forms 
on an evaporated aluminum surface is too thin to furnish sufficient protection, especially 
if the mirror requires frequent cleaning. Silicon monoxide affords a suitable hard 
protective coat. It evaporates at about 1200°C and condenses on the mirror surface 
in uniform and adherent layers. 

There is a considerable difference in the characteristics of coatings when used for 
reflective optics as compared to refractive optics. Figure 8-28 shows the infrared 
reflectance of aluminum protected with a 125-mm film of SiO. 

Figure 8-29 shows the reflectance curve of an "Alzac” reflector. The Alzac reflector 
is an aluminum mirror form whose surface is electrolytically polished or brightened 
and then protected with an anodically produced aluminum oxide film about 1.4 /x thick. 
Figure 8-30 shows the effect of applying a 1.2-/x-thick film of SiO. At 11 /x, where the 
SiO film is effectively one quarter wavelength thick, strong absorption takes place 
and mirror reflectance is decreased to almost zero. 


100 


eg 

w 

u 

z 

H 50 

U 

W 

§ 


WAVELENGTH (p) 


Fig. 8-28. Infrared reflectance of aluminum protected 
with a 125-mp, film of silicon oxide [9], 




Fig. 8-29. Infrared reflectance of polished 
aluminum and of an "Alzac” reflector (alu¬ 
minum electrolytically brightened and 
protected with about 1.4 /x of A1 2 0 3 ) [9]. 



Fig. 8-30. Reflectance of aluminum protected 
with a 1.2-/x-thick film of silicon oxide (SiO 
deposited in 20 min at 1 x 10 -5 mm Hg) [9], 









OPTICAL SURFACE COATINGS 355 

8.6.4. Antireflection Coatings [7,9]. Infrared antireflection coatings increase 

the energy transmitted through optical surfaces by reducing Fresnel reflection losses. 
The criteria for an antireflection coating are ( 1 ) that its refractive index be equal to 
the square root of the substrate index when the substrate is in air: 

n = Vn' ( 8 - 34 ) 

and ( 2 ) that the phase difference between the incident wave and the reflected wave 
be an odd multiple of tt: 

A</> = 477 nd cos r/X = 2n (m + \) 

which reduces to 

nd cos r = (A/4) (2m + 1) (8-35) 

where n = refractive index of film 

n' = refractive index of material 
= phase change of incident light 
X = wavelength of incident light 
d — thickness of film 

r = angle of inclination in the film of the ray to the film normal 
m = 1, 2, 3, . . . 

For more details, see Chapter 7. 

Table 8-16 lists typical antireflection coatings and their effect on the infrared trans¬ 
mission of various materials. The transmittance curves of silicon (Fig. 8-31) and 
germanium (Fig. 8-32) are shown with and without coatings. Figure 8-33 shows the 
measured transmittance curve of an arsenic trisulfide plate coated on both sides with 
double quarter-wave films. Figure 8-34 shows the measured transmittance and re¬ 
flectance of a glass plate coated on each side with a half-wave film of silicon oxide and 
a quarter-wave film of magnesium fluoride. 


Table 8-16. Examples of Antireflection Coating [ 11 ] 


Sample and Thickness 
(in.) 

Fuzed quartz 

0.203 

Dense flint 

0.061 

MgO 

0.129 

AI 2 O 3 

0.063 

AI 2 O 3 

0.126 


U ncoated 
Transmission 

at 2 fx. at 3.6 /x 

0.935 

0.855 

0.86 0.85 

0.88 

0.875 


Coated Transmission 


MgF 2 

at 

2 fx 


LiF 

at 2 ix 

at 3.6 

0.97 

0.97 


0.955 



0.945 


0.94 

0.985 


0.97 


8.6.5. Replica Mirrors [9]. A simple and inexpensive method for producing replicas 
of optical elements, such as mirrors and trihedral prisms, with plastic backings is 
described in reference [9], In this technique replicas equipped with the final high- 
reflecting coatings are prepared directly on the master mold. Epoxy resins are used to 
make the replica mirror form. The epoxy casting compounds are pourable liquids 
that can be hardened overnight with little or no heat through the addition of a liquid 
hardener. Very little heat and very low shrinkage are developed during the hardening 
process. 





EXTERNAL TRANSMITTANCE (%) 


356 


OPTICAL MATERIALS 




Fig. 8-31. Transmittance of a silicon plate, 1.5 mm thick with antireflection 
coatings of: (a) SiO (n = A/4 at 1.7 /Lt ); ( b) ZnS (n — A/4 at 9.8 /a) [12]. 


Fig. 8-32. Transmittance of a germanium 
plate, 1.0 mm thick, with antireflection 
coating of ZnS (n = A/4 at 9.8 /x). 




Fig. 8-33. Transmittance of an As 2 S :! plate 
(n = 2.4) antireflected with double A/4 films 
of tungsten oxide (n — 1.8) and sodium alum¬ 
inum fluoride (n — 1.35), each A/4 thick at 
3 /a [9]. 


Fig. 8-34. Transmittance and 
reflectance of a glass plate with a 
A/2 - A/4 SiO (n = 1.7) and MgF 2 
(n = 1.38) coating, in comparison 
with a A/4 MgF 2 coating [9]. 




























OPTICAL SURFACE COATINGS 


357 



Glass master coated with separating Epoxy resin cast on coated master 
film and final mirror coatings 



Replica separated from master separating layer removed 

Fig. 8-35. The four steps for preparing protected 
replica mirrors with epoxy backing [9]. 


The steps involved in producing replica mirrors with epoxy resins are shown in 
Fig. 8-35. 

In making a replica mirror, the negative glass form is coated with silver, silicon 
monoxide, and a heavy deposit of aluminum. A plastic mold in the form of a ring is 
then placed on the coated glass master and filled with the epoxy resin. After the epoxy 
resin has cured, the replica is separated from the master by applying slight pressure 
to the plastic ring. The silver film, which has poor adherence to glass, is used only 
to make the separation easy. After the separation, the silver film is removed in nitric 
acid, leaving the replica mirror coated with aluminum protected with silicon monoxide. 
The aluminum, which has been deposited rather heavily to produce a rough surface, 
adheres excellently to the hardened epoxy compound. The quality of replica optics 
made with epoxy resins can be greatly improved by adding equal or even higher amounts 
of finely divided silica to the epoxy-type molding compound. The addition of silica 
fillers reduces shrinkage during the hardening process and results in mirror forms 
with increased hardness and heat resistance, and reduced expansion coefficients. On 
an optical test bench, the replica mirror can be compared with the original mirror 
by focusing incident collimated light on pinhole apertures of various sizes and meas¬ 
uring the ratio of the light transmitted by the apertures to the total amount of light in 
the focal plane. The performance of paraboloidal plastic replicas as compared with 
the original glass mirrors is given in Table 8-17. The effects of aging on replica mirrors 
is given in Table 8-18. 


Table 8-17. 

Aperture 

Performance of Plastic Replica Mirrors [13] 

Energy through Aperture (%) 

Size 

(mm) 

Original 
Glass Mirror 

Replica from 
Master No. 1 

Replica from 
Master No. 2 

0.79 

98 

96 

97 

0.38 

96 

89 

90 

0.27 

82 

81 

80 

0.18 

60 

59 

61 

































358 


OPTICAL MATERIALS 


Table 8-18. Effects of Aging on Replica Mirrors [13] 


Aperture 


Energy through Aperture (%) 


Size 

(mm) 

Original 
Glass Mirror 

Replica No. 1 
(2 days old) 

Replica No. 2 
(1 day old) 

Replica No. 2 
(19 months old) 

0.79 

100 

99 

98 

98 

0.38 

99 

97 

98 

98 

0.27 

99 

97 

96 

95 

0.18 

97 

95 

94 

88 


8.7. Radiation Damage 

Protons and electrons inflict damage at a certain depth in a material. This depth 
is a function of the density of the material itself. Thus an absorption cross section 
which has the units g _1 cm 3 is a useful concept, and a penetration "depth” in units of 
g cm -2 is a measure of the susceptibility to damage. Whereas protons and electrons 
damage at a given depth, the number of photons and the resultant damage decreases 
exponentially with distance. Thus photon range is defined as the g cm -2 at which the 
flux is reduced by a factor of e. Figure 8-36 presents the resistance of various materials 
to radiation damage, and Fig. 8-37 is a plot of silicon absorption coefficient as affected 
by neutron bombardment. 



Fig. 8-36. Radiation resistance of various types of materials [14]. 










































OPTICAL PROPERTIES OF BLACKS 


359 


Fig. 8-37. Absorption coefficient 
vs. wavelength for neutron- 
bombarded silicon at room tem¬ 
perature [15]. 


ENERGY (ev) 



Radiation causes ionization and atomic displacement. The rate of ionization can 
be given in erg g~ x yr _1 , and displacement in fractions of atoms displaced per year. 
Tables 8-19 and 8-20 [16] provide general data on the performance of materials. More 
specific data can be found in other references [17 to 70]. 

8.8. Infrared Transmission of Cooled Optical Materials [71] 

Figure 8-38 shows the transmittance of a number of cooled optical materials in the 
1-5.5-p. region. In most cases, the cooled samples transmit more than the uncooled 
samples. If the increase in transmission due to the decrease of Fresnel reflection loss 
at the sample surface is subtracted from the measured total increase of transmission, 
the absolute increase in transmission of the material is obtained. Some of the increased 
transmission results from the reduced path length through the coolant, a function of 
the sample thickness. However, calculations [71] show that this increase can never 
be greater than 0.8% for the thickest sample. 

8.9. Optical Properties of Blacks 

Spectral emittances of blacks have been measured by Stierwalt, Kirk, and Bernstein 
[72]. The data are plotted in Fig. 8-39. Additional data are given by Harris and 
Cuff [73] on specular and diffuse reflectances — including goniophotometric curves —for 
acetylene smoke black, Dupont flat black (unspecified), Eastman Kodak NOD-18, 
Goldblack, Lampblack, soot, and lacquer. All were deposited on glass microscope 
slides and measured from 0.254 to 1.1 /x. 

Other data on the emissivity (emissance) of materials can be found in an Arthur D. 
Little report [74]. They include measurements of Globars, BN, SiC, zirconia, magnesia, 
alumina, aluminum, Inconel, stainless steel; also white enamel and black enamel on 
Inconel, and SiO on platinum and Inconel. The data are not for blacks directly, but 
they may be useful. 





360 


OPTICAL MATERIALS 


Table 8-19. Radiation Dosages Produced by Atomic Particles in Space 

Ionization Fraction of Atoms Displaced 



(erg g- 1 

yr _I ) 


per Year 


Radiation 

Surface 
and Through 

1 mg cm -2 

Through 

1 g cm -2 

Exposed 

Surface 

Through 

1 mg cm -2 

Through 

1 g cm -2 

Inner radiation belt 

10 14 (?) 

10 7 -10 8 

io-»(?) 

10- 5 

10- 9 

Outer radiation belt 

10*3-10 15 

10 6 -10 8 

10 12 -10- 10 

io- i 2 -io - 10 

IO " 13 

Solar emission (flares 
except as noted) 

10 7 -10 9 (?) 

10 4 -10 5 

io- i 2 -io° n 

io- i 2 -io- n 

io- i 3 -io n 

Cosmic rays 

10 2 -10 3 

10 2 -10 3 

10-14-10 13 

io- i 4 -io - 13 

10" 14 -10- 13 


"May be displaced by steady solar emission. 


Table 8-20. Radiation Dose Producing Appreciable Change 
in Engineering Properties of Various Materials 


Material 

Changed Properties 

Ionization 
(erg g- 1 ) 

Fraction of 
Atoms Displaced 

Plastics 

Tetrafluoroethylene 

In air, mechanical, electrical 

10 6 -10 7 

— 


No air, mechanical, electrical 

10 7 -10 9 

— 

Other 

Optical transmission 

10 6 -10 n 

— 


Dimensions, mechanical, electrical" 

10 7 -10 n 

— 

Elastomers 

Mechanical 

10 8 -10 10 

— 

Oils and greases 

Lubrication, consistency 

10 9 -10 12 

— 

Ceramics 

Glass 

Optical transmission 

10 5 -10 10 

10-n.l0-7 


Dimensions, mechanical 

>10 n 

~io - 7 


Electrical 

>10 u 

io- 7 -io - 6 

Fused silica 

Optical transmission 

10 7 -10" 

io- 9 -io - 5 

Crystalline 

Optical transmission 

10 5 -10 n 

10-8-10- 4 


Dimensional, mechanical 

>10 11 

10- 4 -10- 2 


Electrical 6 

>10 n 

10- 3 -10-> 

Semiconductor (devices) 

Minority carrier effects 

— 

10- 12 -10- 10 


Majority carrier effects 

— 

10- 9 -10- 6 

Metals 

F erromagnetism 

— 

10-6-10-5 


Mechanical 

— 

10 4 -10- 3 


Electrical 

— 

10-3-10- 2 


"Temporary increase in electrical conductivity during irradiation at dose rates — 10 s erg g~ l yr _1 . 
"Temporary increase in electrical conductivity during irradiation at dose rates — 10 8 erg g ' yr~'. 




OPTICAL PROPERTIES OF BLACKS 


361 



o 

o 





nr 

~r 

~r 

- 




j 

- 





- 




zS 




_x_ 

_i_ 

Jifc 

_1_ 

_i_ 


o 

o 


as 

H 
Tf o 
Z 
w 

J 

w 

> 


T3 

£ T3 

§ S 

c U 

D °co 
cm 
+ 


8 Z 

V 5 


E 

E 


u 
o 

s •« 

OO ^ 


O 

O 


T3 

0) 


SC 
H 
-r O 
Z 
w 
J 
w 
> 


rt 

u 6 
"o 
H in 
SS 05 

rH 

T3 I 

s 


5 


T3 
6 

=* * 
2 o 

U C£> 
J5 CM 

Z + 





n - 

1 

—r~ 

- 



! 

i 

t 

- 

- 



! 

I 

- 

- 



i 

J 

- 

_i_ 

_L_ 

_L. 



_i_ 

_L_ 


O 

o 


T3 

0> 

T3 

£ 

E 

i§ 

q 

ctJ 


. u 

o 

CO 

* 5 

o 

■T. 

H 3 


w 

o 

CM 

+ 

a> 

2 

Z : 



w • 


o 

J T3 


2 

w 2J 

£ 

H 

> g 

3 



C 

CTJ 

U 


*g o> 



— r 

~r 

~r 



- 

V5r 







- 





V Y» 



- 




- 

i 

_L_ 

\ 

_i_ 


^L_ 


o 

o 


E J 


T3 

V 

j o u ® 
1 O c co 
•> u ctf 0> 

ac c c 


o 

<o ■- 

eg -F 
+ H 


£ 5 05 
^•o • ° ° 


z 
u 

tJ T3 

w ® 

< B 

> o 


~r 

~r 

~r~ 

Jf 

~T~ 

~r 





- 

f 

—-- 



- 

/f 

! 




- 

'T - 

li_ 

_i_ 

_L_ 

_L_ 

_L_ 


lO 

T3 



V 


a. 

ac 

O 

o 

o 

c 

T3 

C 

d 

u 

E-| 

D 

o 

O 

i 

CO 

z 

• 

• 

CM 

+ 

w 



J 

fa 

> 

-a 

2 

C 

o 

„ c 

8 

v*-t 


o 

a> 

H 


J= 

E- 


£ 

E 


* u 
fa o 
■ co 

3S“? 

U- i o 


«) 

W 

<D 

C 

o 


~i 

~T~ 

~r~ 

— r 


- 




- 

- 





-< 




_ 





l 

_i_ 

_L_ 

_l_ 

_i_ 


T3 

<D 


=*■ 8 


x 

H 

rr O 

z 

u 

J 

w 

> 

"S 


E 

E 


o 

o 

s <« 

2 T 

I o 


o 

o 



% ‘aoNvxxmsMVHX 


% ‘30NVXXUMSNVHX 


o 
© 

H 

% ‘aoMvxxmsNVHX 


Ll rj 

a>-V 

-*->o 

^ CO 
00 

8 1 

£? 

.8 o 

£°<o 



C0 

13 

• F-* 

Sh 

o> 

03 

£ 

"c5 

o 

'-3 

a 

o 

T3 

CD 

1 


~r~ 

~1~ 

~T~ 

~r 



r 



- 

- 

v 

Y) 



- 

-ff 

! 




- 

t 

\ 1 

_ 1 1 
Ji 

_i_ 

_1_ 

_i_ 

_i_ 


O 

o 


O 

o 

CO 

00 


*8S 

£ = 

E- 1 D 

a 
z 
w 
•J 

>1 
'So 


? a 

m £ 

o E 

CQ 

« “ 
rt co 
” <u 
© c 

CM .* 

2 sz 
OS E-> 



•Op 
a> o 

^ in 

8 2 

c 1 

=> ■§ 

d 


u 

o 

T3 CO 
0> CM 

8 I 

o 


1 > o ™ 

^ I 1-1 

as 
5 



o 

O 

CO 

-S 22 


T3 
1 C 



<© T3 
<D 

*8 

eg « g d 

n 3 s: ^ c 

•& O i rt c 

Z ; .£ O fi . 

E % <=>. ' 

_3 N 

’rt + co 
. -S » 

E rt « 
.2 
a 

rt a> .e 
U fa E- 




fa . 

■J 

W "O 

> -I 
« 8 
£ o 


O 

O 


"1 TT 


o 

o 





i 



— r 

— r 

T 

- 





- 

- 





- 


r 

'..i* 

_ 



- 



L 

_i_ 

_x_ 

_L_ 


o 

o 


o 

o 




o 

o 


T3 

O 


X 
H 

a 
z 
w 

<8 

^o 


W 

W 

D 

C 

o 

2 

H 


% ‘aoMvxxuvsNVHX 


% ‘aoMvxxmsNVHX 


% ‘30MVXXIWSNVHX 
























































































































































































































362 


OPTICAL MATERIALS 



Fig. 8-39. Specular emittance of some useful "blacks;” the samples are from Barnes 
Engineering Co. and identified both by their Barnes label and their generic name: 

1. Pryomark Standard Black (BEC-1); 2. Sicon Black (BEC-2); 3. Krylon Flat Black 
(BEC-3); 4. Magnesium oxide powder (BEC-4); 5. Iron oxide. 

8.10. Optical Properties of Water 

Water in many forms —vapor, liquid, solid, distilled, brine, etc.— has a number of 
applications in the infrared. Some of the useful and interesting properties are given 
here. Normal incidence values of specular reflectivity and values of n and k are 
given by Centeno [75] are given in Table 8-21; Fig. 8-40 gives the reflection and Fig. 8-41 
the refractive index. Values for different angles of incidence are given by McSwain 
and Bernstein [76]. Similar data are given by Kislovskii [77]; Ockman [78] gives 
information on ice. 



Fig. 8-40. Specular reflectance of water as a function of incident 
angle and wavelength. 



Fig. 8-41. Index of refraction of water calculated from reflectivity. 











OPTICAL PROPERTIES OF WATER 


363 


Table 8-21. Optical Constants of Water in the Near Infrared [75] 


Wavelength 

Reflectivity 

Extinction 

Refractive 

Wavelength 

Reflectivity 

Extinction 

Refractive 

(/x) 

(%) 

Coefficient 

Index 

(m) 

(%) 

Coefficient 

Index 

1.00 

1.96 

3.5 X 10 « 

1.326 

5.85 

1.25 

0.0653 

1.242 

1.05 

1.95 

3.1 

1.325 

5.90 

1.50 

0.0799 

1.266 

1.10 

1.94 

1.7 

1.324 

6.00 

2.02 

0.1220 

1.304 

1.20 

1.93 

1.21 X 10-* 

1.323 

6.04 

2.16 

0.1372 

1.312 

1.30 

1.91 

1.29 

1.321 

6.10 

2.28 

0.1216 

1.331 

1.40 

1.90 

1.30 X 10 « 

1.320 

6.20 

2.46 

0.0940 

1.358 

1.50 

1.88 

3.12 

1.318 

6.30 

2.34 

0.0768 

1.352 

1.60 

1.87 

1.12 

1.316 

6.40 

2.22 

0.0626 

1.344 

1.70 

1.85 

1.08 

1.315 

6.50 

2.12 

0.0524 

1.336 

1.80 

1.82 

1.43 

1.312 

6.60 

2.05 

0.0462 

1.331 

1.90 

1.79 

8.14 

1.309 

6.70 

2.00 

0.0450 

1.326 

2.00 

1.74 

14.3 

1.304 

6.80 

1.98 

0.0448 

1.324 

2.20 

1.63 

5.11 

1.293 

6.90 

1.97 

0.0451 

1.323 

2.40 

1.47 

10.1 

1.276 

7.00 

1.95 

0.0457 

1.321 

2.50 

1.37 

20.1 

1.265 

7.20 

1.87 

0.0463 

1.313 

2.60 

1.25 

51.8 

1.252 

7.40 

1.77 

0.0466 

1.303 

2.70 

0.96 

0.0183 

1.216 

7.60 

1.72 

0.0484 

1.298 

2.74 

0.75 

0.0273 

1.187 

7.80 

1.68 

0.0472 

1.294 

2.77 

0.90 

0.0364 

1.206 

8.00 

1.66 

0.0472 

1.292 

2.80 

1.41 

0.0490 

1.266 

8.20 

1.64 

0.0472 

1.289 

2.85 

2.00 

0.0578 

1.324 

8.40 

1.62 

0.0473 

1.287 

2.90 

2.48 

0.0637 

1.367 

8.60 

1.57 

0.0476 

1.282 

2.95 

2.87 

0.0670 

1.401 

8.80 

1.51 

0.0483 

1.276 

3.00 

3.40 

0.0680 

1.446 

9.00 

1.44 

0.0498 

1.268 

3.02 

3.90 

0.0683 

1.453 

9.2 

1.35 

0.0518 

1.257 

3.07 

4.39 

0.0682 

1.525 

9.4 

1.27 

0.0542 

1.247 

3.10 

4.30 

0.0681 

1.517 

9.6 

1.18 

0.0562 

1.236 

3.16 

4.12 

0.0658 

1.504 

9.8 

1.09 

0.0595 

1.224 

3.20 

4.00 

0.0611 

1.495 

10.0 

0.99 

0.0601 

1.212 

3.30 

3.65 

0.0370 

1.471 

10.2 

0.92 

0.0621 

1.202 

3.40 

3.37 

0.0225 

1.449 

10.4 

0.85 

0.0687 

1.190 

3.50 

3.05 

0.0128 

1.423 

10.6 

0.78 

0.0802 

1.175 

3.60 

2.80 

0.00783 

1.402 

10.8 

0.73 

0.0946 

1.159 

3.70 

2.59 

0.00619 

1.381 

10.9 

0.70 

0.0993 

1.150 

3.80 

2.36 

0 00596 

1.363 

11.0 

0.72 

0.1138 

1.143 

3.90 

2.25 

0.00590 

1.353 

11.1 

0.77 

0.1290 

1.137 

4.03 

2.19 

0.00642 

1.347 

11.2 

0.84 

0.1471 

1.129 

4.10 

2.18 

0.00709 

1.346 

11.3 

0.92 

0.1637 

1.121 

4.20 

2.16 

0.00871 

1.344 

11.4 

0.97 

0.1751 

1.114 

4.30 

2.15 

0.0108 

1.343 

11.5 

1.02 

0.1831 

1.111 

4.40 

2.14 

0.0133 

1.342 

11.6 

1.09 

0.1880 

1.118 

4.50 

2.13 

0.0164 

1.341 

11.7 

1.19 

0.1937 

1.130 

4.60 

2.11 

0.0194 

1.339 

11.8 

1.30 

0.1991 

1.144 

4.70 

2.10 

0.0202 

1.338 

12.0 

1.47 

0.2058 

1.165 

4.80 

2.08 

0.0199 

1.336 

12.5 

2.15 

0.2438 

1.219 

4.90 

2.05 

0.0188 

1.334 

13.0 

3.02 

0.2918 

1.270 

5.00 

2.02 

0.0169 

1.331 

13.5 

3.55 

0.3202 

1.297 

5.10 

1.98 

0.0152 

1.327 

14.0 

4.10 

0.3582 

1.309 

5.20 

1.93 

0.0143 

1.322 

14.5 

4.72 

0.4028 

1.313 

5.30 

1.86 

0.0143 

1.315 

15.0 

5.12 

0.4298 

1.315 

5.40 

1.80 

0.0146 

1.309 

16.0 

5.29 

0.4088 

1.374 

5.50 

1.73 

0.0174 

1.302 

17.0 

6.15 

0.4070 

1.468 

5.60 

1.66 

0.0232 

1.295 

18.0 

6.21 

0.4585 

1.401 

5.70 

1.57 

0.0340 

1.284 





5.80 

1.40 

0.0515 

1.263 






364 OPTICAL MATERIALS 

8.11. Effects of Space Radiation on Optical Materials 

The large flux and energy of the electrons in the low-lying Van Allen belt indicates 
that effects on the spectral transmission of various materials may be appreciable. 
An example of the effect of 1-Mev electrons on the transmission of various materials 
is shown in Table 8-22, taken from data by Cooley and Janda [79]. Many of the mate¬ 
rials shown in Table 8-22 are also used as cover materials for solar cells in satellites. 

The decrease in transmission of fused quartz windows as a result of radiation occurs 
fairly rapidly during the initial exposure, with the change being much less in mag¬ 
nitude as the total radiation dose accumulates. There have also been large differences 
observed in the amount of reduction in transmission as a result of radiation, and this 
is believed to be due to the impurity content of the quartz. In the studies of solar 
cell windows [79] the poorest quartz had 50 to 70 ppm of impurities, and the best had 
less than 1 ppm. From these studies it was apparent that extreme care is essential 
in the selection of transparent material such as fused quartz if a minimum of reduced 
transmission as a result of radiation is desired. 

In the case of glasses and crystals used as optical elements in infrared systems, 
the effects of radiation have not been extensively studied. In the case of glasses it 
would be expected that color formation under a radiation environment would be severe. 
As a result of "built in” impurity centers in the glass network, radiation may form 
displaced electrons with negative ion sites or vacancies, to form color centers. It ap¬ 
pears, however, that although color centers severely affect the visible transmission, in 
some cases they do not appreciably influence the transmission characteristics in the in¬ 
frared region. Until more information is available it appears that the problems for a 
given optical element might best be solved by exposing this element to radiation levels 
corresponding with the flux and energy expected to be present in the particular orbit 
encountered in its space environment. The element should be examined before and 
after radiation in the desired spectral region. 


Table 8-22. Effects of 1-Mev Electrons on Spectral Transmittance 
of Optical Materials (Total Dose of 10 16 e ! cm 2 ) 




50% Cutoff 
Point (p) 

Wavelength (p) 


Material Description 


0.40 

0.50 

0.60 

0.70 

1. Microsheet, Corning 0211, 6 mil 

To 

_ 

89.0 

90.0 

90.0 

90.0 


77 

— 

82.0 

85.0 

87.0 

99.0 


A T 

- 

7.9 

5.6 

3.3 

2.2 

2. Fused silica. Coming 7940, 

To 

— 

89.0 

90.0 

90.0 

90.0 

66 mil 

77 

— 

88.5 

88.5 

88.0 

89.0 


A T 

- 

0.6 

1.7 

2.2 

1.1 

3. Fused silica, Coming 7940, 

20 mil + antireflecting coating 

To 

0.416 

Below 50% 
cutoff 

89.0 

90.0 

91.0 

+ "blue” filter 

7V 

0.425 

Below 50% 
cutoff 

88.0 

88.0 

90.0 


A T 

- 

- 

1.1 

2.2 

1.1 

4. Nonbrowning lime glass 

To 

0.365 

89.0 

89.0 

89.0 

89.0 

Coming 8365, density 2.7, 

77 

— 

29.0 

30 

33 

35 

60 mil* 

A T 

- 

- 

- 

- 

— 

5. Nonbrowning lead glass 

To 

0.380 

72.0 

73.0 

74.0 

74.0 

Coming 8365, density 3.3, 

77 

— 

21.0 

22.0 

22.0 

24.0 

60 mil 

AT 

- 

- 

— 

- 

— 

6. High-density lead glass 

To 

0.392 

82.0 

84.0 

85.0 

84.0 


T r 

0.402 

72.0 

78.0 

82.0 

83.0 


AT” 

— 

12.2 

7.1 

3.5 

1.2 


T 0 = Initial preirradiated transmittance. 

TV = Transmittance after 10 18 e/cm 2 . 

A T = Decrease in transmittance. 

^Radiation caused internal crazing in specimen diffusing light. 




REFERENCES 


365 


References 

1. S. S. Ballard, K. A. McCarthy, and W. L. Wolfe, Optical Materials for Infrared Instrumentation, 
2389-11-S (1959); Supplement 2389-11-Si (1961), The University of Michigan, Ann Arbor. 

2. J. Jerger, Jr., Investigation of Long Wavelength Infrared Transmitting Glasses, Tech. Docu¬ 
ment Report ASD-TDR-63-552, Servo Corp. of America, Hicksville, N.Y. (1963). 

3. New High Temperature Infrared Transmitting Glasses, Texas Instruments Incorporated, 
Dallas, Texas (1963). 

4. M. Herzberger and C. D. Salzberg, J. Opt. Soc. Am., 52, 420 (1962). 

5. Quarterly Progress Report, July, August and September 1963, Tech. Rept. No. 2900-455-P, 
Institute of Science and Technology, The University of Michigan, Ann Arbor, Mich. (Nov 1963). 
CONFIDENTIAL 

6. A. E. Conrady, Applied Optics and Optical Design, Dover, New York (1957). 

7. O. S. Heavens, Optical Properties of Thin Solid Films, Butterworth Scientific Publications, 
London (1955). 

8. H. O. McMahon, J. Opt. Soc. Am., 40, 376 (1950). 

9. G. Hass and A. F. Turner, "Coatings for Infrared Optics,” Reprint from Wissenschaftliche 
Verlagsgesellschaft m. b. h., Stuttgart, 143-163. 

10. Private Communication, Haynes Stellite Co., Kokomo, Ind. 

11. J. R. Jenness, Jr., J. Opt. Soc. Am., 46, 157 (1956). 

12. J. T. Cox and G. Hass, J. Opt. Soc. Am., 48, 677 (1958). 

13. A. P. Bradford, W. W. Erbe, and G. Hass, J. Opt. Soc. Am., 49, 990-991 (1959). 

14. G. R. Hennig, Moderator, "Shields and Auxiliary Equipment,” talk given at the colloquium 
on the Effect of Radiation on Materials at Johns Hopkins University, Baltimore, Md. (1957). 

15. T. A. Longo, Nucleon Irradiation of Silicon Semiconductors, Special Report to Signal Corps., 
Purdue University, LaFayette, Ind. (1957). 

16. L. D. Jaffe and J. B. Rittenhouse, Behavior of Materials in Space Environments, Technical 
Report 32-150, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 
Calif. (1961). 

17. R. W. King, N. J. Broadway, and S. Palinchak, The Effect of Nuclear Radiation on Elastomeric 
and Plastic Components and Materials, REIC Report No. 21, The Radiation Effects Information 
Center, Battelle Memorial Inst., Columbus, Ohio (1961). 

18. K. Kobayaski, Phys. Rev., 102, 348 (1956). 

19. A. F. Cohen and L. C. Templeton, Solid State Division Progress Report, ORNL-2188, Oak 
Ridge National Laboratories, Oak Ridge, Tenn. (1956). 

20. R. Berman, F. E. Simon, P. G. Klemens, and T. M. Fry, Nature, 166, 277 (1950). 

21. J. J. Harwood, H. H. Hausner, and J. G. Morse, The Effects of Radiation on Materials, Reinhold, 
New York (1958). 

22. F. A. Bovey, Effects of Ionizing Radiation on Natural and Synthetic High Polymers, Inter¬ 
science, New York (1958). 

23. N. J. Broadway, M. A. Youtz, S. Palinchak, and R. Mayer, Effect of Nuclear Radiation on 
Elastometer and Plastic Materials, Battelle Memorial Institute, Radiation Effects Information 
Center, Rept. 3 and Addenda, Columbus, Ohio (1958-1960). 

24. J. F. Fowler and F. T. Farmer, Nature, 171, 1020-1021 (1953). 

25. J. F. Fowler and F. T. Farmer, Nature, 174, 136-137 (1954). 

26. L. A. Wall and R. E. Florin, J. Appl. Polymer. Sci. 1, 251 (1959). 

27. C. D. Bopp and O. Sisman, Nucleonics, 13, No. 7, 28-33 (1955). 

28. C. D. Bopp and O. Sisman, Nucleonics, 13, No. 10, 51-55 (1955). 

29. R. A. Mayer, N. J. Broadway, and S. Palinchuk, Effect of Nuclear Radiation on Protective 
Coatings, Battelle Memorial Institute, Radiation Effects Information Center, Rept. 13, Colum¬ 
bus, Ohio (1960). 

30. J. P. Shoffner, J. of Teflon, 2, No. 1, 6-7 (1961). 

31. J. H. Coleman and D. Bohm, J. Appl. Phys., 24, 497-498 (1953). 

32. A. J. Warner, F. A. Muller, and H. G. Nordlin, J. Appl. Phys., 25, 131 (1954). 

33. P. Y. Feng and J. W. Kennedy, J. Am. Chem. Soc., 77, Part 1, 847-851 (1955). 

34. A. Charlesby, "Effects of Radiation on Behavior and Properties of Polymers,” in J. J. Harwood, 
H. Hausner, J. G. Morse, and W. G. Rauch (eds.), Effects of Radiation on Materials, Reinhold, 
New York, 261-286 (1958). 

35. K. R. Ferguson, "Design and Construction of Shielding Windows,” Nucleonics 10, No. 11, 
46-51 (1952). 

36. G. S. Monk, "Coloration of Optical Glass by High-Energy Radiation,” Nucleonics 10, No. 11, 
52-55 (1952). 


366 


OPTICAL MATERIALS 


37. N. J. Kreidl and J. R. Hensler, J. Am. Ceram. Soc., 38, 423-32 (1955). 

38. N. J. Kreidl and J. R. Hensler, J. Opt. Soc. Am., 47, 73-74 (1957). 

39. R. L. Hines, J. Appl. Phys., 28, 587-91 (1957). 

40. J. F. Hansen, S. E. Harrison, W. L. Hood, D. J. Hamman, W. E. Chapin, and E. N. Wyler, 
Effect of Nuclear Radiation on Electronic Components and Systems, Battelle Memorial Institute, 
Radiation Effects Information Center, Repts. 2 and 12, Columbus, Ohio (1957 and 1960). 

41. J. K. Davis, Electrical Mfg. 59, No. 6, 151-156 (1957). 

42. M. Levy and J. H. O. Varley, Proc. Phys. Soc., B68, 223-233 (1955). 

43. A. J. Gale and F. A. Bickford, Nucleonics 11, No. 8, 48 (1953). 

44. F. S. Dainton and J. Rowbottom, Trans. Faraday Soc., 50, 480-493 (1954). 

45. P. W. Levy, J. Chem. Phys., 23, 764-765 (1955). 

46. W. Primak, L. H. Fuchs, and P. Day, J. Am. Ceram. Soc., 38, 135-139 (1955). 

47. J. H. Crawford, Jr., and M. C. Wittels, "Review of Investigations of Radiation Effects in Ionic 
and Covalent Crystals,” Proc. Inti. Conf. on Peaceful Uses of Atomic Energy, United Nations, 
N.Y. (1955), 7, Nuclear Chemistry and Effects of Irradiation, 654-665. 

48. M. Swerdlow and R. F. Geller, Survey of Radiation-Resistant Glass, U.S. National Bureau 
of Standards unpublished report to U.S. Air Force, Wright Air Development Center, on Order 
33(616)56-17 (1957). 

49. L. Reiffel, R. Estin, D. Kazen, L. Marchi, and H. Nakamura, Radiation Sensitive Glasses, 
Armour Research Foundation, Final Rept. to U.S. Army Evans Signal Lab., Project A031-4, 
Chicago (1953). 

50. C. M. Nelson and R. A. Weeks, J. Appl. Phys., 32, 883-886 (1961). 

51. E. W. J. Mitchell and E. G. S. Paige, Phil. Mag., Series 8, 1, 1085-1115 (1956). 

52. G. J. Dienes and G. H. Vineyard, Radiation Effects in Solids, Interscience, New York (1957). 

53. F. Seitz and J. S. Koehler, Displacement of Atoms During Irradiation, in F. Seitz and D. Turn- 
bull (eds.), Academic Press, New York, 2, 307-448 (1956). 

54. D. S. Billington, "Radiation Effects in Metals and Alloys,” in Effects of Radiation on Materials, 
J. J. Harwood, H. Hausner, J. G. Morse, and W. G. Rauch (eds.), Reinhold, New York, 99-125 
(1958). 

55. B. C. Allen, A. K. Wolff, A. R. Elsea, and P. D. Frost, Effect of Nuclear Radiation on Structural 
Metals, Battelle Memorial Inst. Radiation Effects Information Center, Rept. 5, Columbus, 
Ohio (1958). 

56. P. J. Reid, J. W. Moody, Effect of Nuclear Radiation on Magnetic Materials, Battelle Memorial 
Inst. Radiation Effects Information Center, Tech. Memo 12, Columbus, Ohio (1958). 

57. J. M. Denney, "Radiation Damage to Satellite Solar Cell Power Systems.” Am. Rocket Soc. 
Reprint 1295-60 (1960); in "Energy Conversion for Space Power,” ed. by N. W. Snyder, Aca¬ 
demic Press, New York, 345-61 (1961). 

58. H. Y. Fan and K. Lark-horovitz, "Irradiation Effects in Semiconductors,” in Effects of Radiation 
on Materials, J. J. Harwood, H. Hausner, J. G. Morse, and W. G. Rauch (eds.), Reinhold, New 
York, 99-125 (1958). 

59. C. M. Nelson and R. A. Weeks, J. Appl. Phys., 32, 883-886 (1961). 

60. J. C. Pigg and C. G. Robinson, Electrical Mfg., 59, No. 4, 116-124 (1957). 

61. G. Enslow, F. Junga, and W. W. Happ, Gamma Radiation Effects in Silicon Solar Cells, in 
3rd ARDC Radiation Effects Symp., 4, Lockheed Nuclear Products Div., Marietta, Georgia 
(1958). 

62. G. L. Keister and H. V. Steward, Proc. Inst. Radio Eng., 45, 931-937 (1957). 

63. H. L. Steele, "Effects of Gamma Radiation on Transistor Parameters,” Proc. Transistor Re¬ 
liability Symp., New York University, New York, 96 (1956). 

64. J. E. Drennen and S. E. Harrison, Effect of Nuclear Radiation on Semiconductor Diodes, Battelle 
Memorial Institute, Radiation Effects Information Center Tech. Memo 6, Columbus, Ohio 
(1958). 

65. F. Gordon, Effects of Nuclear Radiation on Power Transistors, in 3rd ARDC Radiation Effects 
Symposium, 4, Lockheed Aircraft Corp., Nuclear Products Div., Marietta, Georgia (1958). 

66. T. E. Lusk, Electronic Design, 8, No. 22, 74-75 (1960). 

67. R. G. Downing, "Electron Bombardment of Silicon Solar Cells,” A. Rocket Soc. Reprint 1294-60 
(1960); Energy Conversion for Space Power, N. W. Snyder (ed.), Academic Press, New York, 
325-344 (1961). 

68. D. S. Billington, "Radiation Effects in Metals and Alloys,” Effects of Radiation on Materials, 
J. J. Hardwood, H. Hausner, J. G. Morse, and W. G. Rauch (eds.), Reinhold, New York, 99-125 
(1958). 


REFERENCES 


367 


69. B. C. Allen, A. K. Wolff, A. R. Elsea, and P. D. Frost, Effect of Nuclear Radiation on Structural 
Metals, Battelle Memorial Inst. Radiation Effects Information Center, Rept. 5, Columbus, 
Ohio (1958). 

70. F. J. Reid and J. W. Moody, Effect of Nuclear Radiation on Magnetic Materials, Battelle Memo¬ 
rial Inst., Radiation Effects Information Center, Tech. Memo 12, Columbus, Ohio (1958). 

71. G. F. Linsteadt and H. P. Leet, Proc. IRIS, 6, No. 3, 159 (1961). 

72. D. L. Stierwalt, D. D. Kirk, and J. B. Bernstein, Foundational Research Projects, July-Septem- 
ber 1962, Navweps Report 7237, Naval Ordnance Laboratory, Corona, Calif. (1962). Also in 
in Appl. Optics, 2, 1169 (1963). 

73. L. Harris and K. F. Cuff, J. Opt. Soc. Am., 46, 160 (1956). 

74. Henry H. Blau, Jr., Eleanor Chaffee, John R. Jasperse, and William S. Martin, High Tempera¬ 
ture Thermal Radiation Properties of Solid Materials, Arthur D. Little, Inc., Cambridge, 
Mass. (1960). 

75. M. Centeno, J. Opt. Soc. Am., 31, 245 (1941). 

76. B. McSwain and J. Bernstein, Foundational Research Projects, October-December 1960, 
NOLC 539, Naval Ordnance Laboratory, Corona, Calif. (1961). 

77. L. D. Kislovskii, Optics and Spectroscopy, 7, 201 (1959). 

78. N. Ockman, Dissertation, University of Michigan, Ann Arbor (1957). 

79. W. C. Cooley and R. J. Janda, Handbook of Space Radiation Effects on Solar Cell Power Sys¬ 
tems, NASA Rept. SP-3003 (1963). 



































































































Chapter 9 
OPTICS 


Warren J. Smith 

Infrared Industries, Inc. 


CONTENTS 


9.1. Terminology, Symbols, and Sign Convention. 371 

9.1.1. Terminology. 371 

9.1.2. Symbols. 371 

9.1.3. Sign Convention. 371 

9.2. First-Order (Gaussian) Optics. 371 

9.2.1. Focal Points and Principal Points. 371 

9.2.2. The Cardinal Points of Elements. 373 

9.2.3. Multielement Optical Systems. 375 

9.2.4. Paraxial Ray Trace — Surface by Surface. 377 

9.3. Limitation of Rays by Stops and Apertures. 377 

9.3.1. The Aperture Stop. 377 

9.3.2. ' The Field Stop. 378 

9.3.3. Vignetting. 378 

9.3.4. Glare Stops and Baffling. 379 

9.3.5. Relative Aperture, Speed, F/No, 

Numerical Aperture. 379 

9.3.6. Depth of Field and Depth of Focus. 380 

9.4. Aberrations. 381 

9.4.1. Aberration Description. 381 

9.4.2. The Seidel Aberrations. 382 

9.4.3. Correction of Aberrations. 385 

9.4.4. Variations of Aberrations with 

Aperture and Image Size. 385 

9.4.5. Zonal and Residual Aberrations. 386 

9.4.6. Chromatic Variation of Aberrations. 386 

9.4.7. Graphical Representation of Aberrations. 386 

9.5. Ray Tracing. 388 

9.5.1. Ray-Tracing Precision. 388 

9.5.2. Determination of Specific Aberrations. 388 

9.5.3. Ray-Tracing Equations. 390 

9.5.4. Graphical Ray Tracing. 398 


369 

































OPTICS 


9.6. Third-Order Aberrations. 399 

9.6.1. Third- and Higher-Order Aberrations — 

Surface Contributions. 399 

9.6.2. Third-Order Aberration Contributions of Thin Lenses. 402 

9.6.3. Stop Shift Theory. 404 

9.6.4. Afocal Systems. 406 

9.7. Optical Design Techniques. 406 

9.7.1. General Considerations. 406 

9.7.2. Correction of Primary Aberrations. 406 

9.7.3. Reduction of Residual Aberrations. 407 

9.7.4. Automatic Design. 408 

9.8. Achromatism and Achromats. 409 

9.9. Resolution, Definition, and Image Spot Size. 410 

9.9.1. The Airy Disc. 410 

9.9.2. The Effects of Aberrations on the Airy Disc. 413 

9.9.3. Geometrical Limits on Resolution. 413 

9.10. Summary of Equations. 417 


370 


















9. Optics 


9.1. Terminology, Symbols, and Sign Conventions 

The formulas of geometrical optics are the result of selective application of Snell’s law, 

n sin I' = n sin I (9-1) 

This can also be written in vector form. Let Q 0 be a vector representing the incident 
ray and Qi the emergent ray, and let N = the surface normal. Then Snell’s law is 

(n 0 Qo — ftiQi) X N = 0 (9-2) 

Optical systems are usually composed of a series of surfaces of revolution, often spher¬ 
ical. The optical axis of a lens or system is that line upon which the centers of curvature 
of the surfaces lie. Unless the contrary is specifically noted, optical systems are 
assumed to be axially symmetrical. 

9.1.1. Terminology. 

An element is a single piece of optical material ( e.g ., a simple lens or mirror). 

A component is one or more elements treated as a unit {e.g., a cemented doublet). 

A member consists of all the components either ahead of (the front member) or behind 
(the rear member) the aperture stop. 

9.1.2. Symbols. In general, lower-case symbols are used for the paraxial (see 
Sec. 9.2) values of quantities and capitals, for values other than paraxial. Primed 
symbols refer to quantities after refraction by a surface or a lens. Subscripts identify 
the surface or lens with which a quantity is associated. 

9.1.3. Sign Convention. With the exception of certain aberrations, the sign con¬ 
vention follows that given in Applied Optics and Optical Design , by A. E. Conrady, 
Dover, New York (1957). Light is assumed to progress from left to right. Radii and 
curvatures are positive if the center of curvature is to the right of the surface. Surfaces 
or elements are positive if they converge light. Points lying to the right of a point, 
element, or surface are considered to be a positive distance away. Slope angles are 
positive if the ray must be rotated counterclockwise to reach the axis. Angles of 
incidence and refraction are positive if the ray must be rotated clockwise to reach the 
normal to the surface. 

9.2. First-Order (Gaussian) Optics 

The first-order expressions are derived by reducing the exact trigonometrical ex¬ 
pressions for ray paths to the limit when the angles and ray heights involved approach 
zero. These expressions, however, are completely accurate for the paraxial region. 

9.2.1. Focal Points and Principal Points. A well-corrected optical system can 
be treated as a "black box” whose characteristics are defined by its cardinal points, 
which are its first and second focal points and its first and second principal points. 

The cardinal points are shown in Fig. 9-1. 


371 


372 


OPTICS 


Optical 

System 

/ 



Point 

Fig. 9-1. Location of focal and principal points of an optical system. 


9.2.I.I. Image Position. The location of the image formed by an optical system can 
be determined by either of the following expressions: 

i = - + 7; /■* = -**' (9-3) 

s s f 

(the quantities are defined in Fig. 9-2). If the object or image lies to the left of the 
principal or focal point, s, s', x, and x are negative. 


Optical 

System 



Fig. 9-2. Relationships between cardinal points and 
position and size of object and image. 







































FIRST-ORDER (GAUSSIAN) OPTICS 373 


9.2.1.2. Magnification. 
and is given by 


Lateral magnification relates the heights of image and object 
m = — fix = — x' / f = h'/h = s' Is (9-4) 


(the quantities are defined in Fig. 9-2). 

Longitudinal magnification relates the lengths (or depths) along the optical axis of 
image and object and is given by 


m = s'is' 2 /sis 2 = m 2 (approx) (9-5) 

where Si is the distance from the principal points to the edges of the object and image 
nearest the optical system, and s 2 is the distance from the principal points to the edges 
most distant from the optical system. 

9.2.I.3. The Lagrangian Invariant. Figure 9-3 shows an axial ray and a principal 
ray passing through an image-forming optical system. The Lagrangian invariant 
applies wherever either ray crosses the axis. 

/ = hnu — h'n u' = ynu pr (9-6) 


Outside the paraxial region the invariant takes the form hn (sin u ), and is also called 
the Law of Sines. A rearrangement of Eq. (9-6) yields an additional relationship 
for the magnification: 

n nu u .. 

m = — = = — (in air) (9-7) 

h n u u 


Optical System 



9.2.2. The Cardinal Points of Elements. Figure 9-4 shows the location of the car¬ 
dinal points of different types of elements. Their values are (see Fig. 9-4a for definitions 

of symbols). _ nr x r 2 convergent f > 0 

' ' A divergent f < 0 


8 = 




d — (n — 1) ^ , 


<£' = (n — 1) -f- 

A 


where A = (n — l)[n(r t — r 2 ) — (n — 1)0 



r i 

n — 1 ’ 



nr\ 
n — 1 




r 2 

1 — n 


















374 


OPTICS 


Positive (converging) 


Negative (diverging) 




Meniscus 



Concave Mirror 




Meniscus 



Convex Mirror 



Fig. 9-4. Location of principal points for thick 
lenses of various common shapes. 



Fig. 9-4a. Definitions of symbols. 





















































FIRST-ORDER (GAUSSIAN) OPTICS 


375 


9.2.2.I. Thick-Lens Equations. The cardinal points of an element of appreciable 
thickness are given by the following equations: 


power = </>=— = (n — 1) 


=(n — 1) 


1 1 | (n - l)t 

ITi r 2 nr x r 2 


, (n - 1 ) 

Ci — c 2 H - tc x c 2 

n 


(9-8) 


back focus = b.f.l. = f— —-— tf 


nr i 


(9-9) 


9.2.2.2. Thin-Lens Equations. A thin lens can be defined as a lens whose thickness 
is negligible compared to the accuracy required for the calculation. If one sets t— 0, 
Eq. (9-8) reduces to 


(/> =~ = (n - 1) (jr — = (n - l)(c, - c 2 ) (9-10) 

The power of a thin element can be held constant for an infinite number of element 
shapes by varying the ratio of the curvatures of radii while (ci — c 2 ) is held constant, 
a process called bending. 

The principal points of a thin lens are taken to be coincident with its location. 

9.2.2.3. Mirrors. The cardinal points of a mirror in air may be derived by assuming 
an index of (—1) for the medium following the reflecting surface and applying the thin- 
lens expression for power, neglecting c 2 . Thus, for a mirror, 


4>=7=(-l-l)- = —= -2 c 
/ r r 

f = — 0.5r 


(9-11) 


The principal points are coincident with the mirror surface. 

The sign convention for distance after reflection is reversed as a result of the re¬ 
flection index of (—1). Thus, the distance from the mirror to an image lying to the left 
of the mirror is taken as positive instead of negative. 

9.2.3. Multielement Optical Systems. For optical systems of more than two 
elements, it is preferable to determine the image locations and sizes, and the cardinal 
points, by tracing the paths of specific rays. It is also possible to calculate the position 
and size of an image by repeated application of Eqs. (9-3), (9-4), and (9-5). 

9.2.3.I. Ray Tracing a Series of Elements. Given a series of elements with powers 
4> i, <f> 2 , <f> 3 , ...,</>„ separated by distances d\, d' 2 , d' 3} ..., d'„-i, the path of a ray through 
the series may be traced by repeated applications of the following equations: 

u 'i — Ui + ynfn (9-12) 

y i + i = y, - d'iUi (9-13) 


where u t is the slope of the ray approaching the element i 

u'i is the slope of the ray it has passed after through element i 

y, is the height at which the ray strikes the element 

(These equations are most useful for tracing through a series of thin lenses, but they 
can be applied equally well to thick lenses by taking y as the height at which the ray 
strikes the principal planes and d as the spacing between principal points.) 










376 


OPTICS 


Thus, to determine the cardinal points of a system, one can trace a ray with u x = 0 
and a nominal value for yi. Then the effective focal length is given by 

f=yju' n (9-14) 

and the back focal length is given by 

b — y n l u' n (9-15) 

Reversing the system and repeating the process will yield the other set of cardinal 
points. 

Alternatively, a ray may be traced from the axial intercept of the object through 
the system. The starting slope for such a ray is given by 

«i = yi/si (9-16) 

and the image is located at 

s'n = yju' n (9-17) 

The size of the image is then 

h' = huju'n (9-18) 

Also, a principal ray may be traced from a specific off-axis point in the object; its intercept 
in the image plane is then the image of the object point. This height may thus be found 
from 

h’ — y pr — s'u'pr (9-19) 

where the subscript pr indicates the principal-ray data. 

9.2.3.2. Combination of Two Elements. Given two elements, A and B, separated by 
a distance d, with focal lengths f A and f B , powers <f A and 4>b, a combined focal length 
f AB , and back focus s', and v the distance from element A to the focus {v = d + s'), then 
one or more of the following expressions will be found to cover almost any desired 
relationship. 


fAB 


f A fB 

fA "f" fB d 


(9-20) 


fs(fA-d) {f a — d) 

s — ~ : ~ ; — f ab 7 — Jab\ 1 — 4> A d) 


fA + fB — d 


fA 


fABifA -V) (V- f A ) + V(U ~ fA) 2 ~ 4 f B {V ~ f A ) 

fA ~ fAB 2 


fs - 


f A Bd 
fAB ~ S' 

—s'd 

fAB - s' — d 


j _ f bs' __.f_.f_ f A f H _ ( v + A) + V (v ~ fA) 2 — 4f B {v 
f B ~ s' ' A Tb f AB 2 

If a minimum £</> is desired where v and f AB are given, one sets 


~fA) 


(9-21) 

(9-22) 

(9-23) 

(9-24) 


and 


d — V f A B — "V/ f AB^f ab v ) 

S' = f AB ± fA B (fA B ~ V) 


(9-25) 

(9-26) 
















LIMITATION OF RAYS BY STOPS AND APERTURES 


377 


9.2.4. Paraxial Ray Trace — Surface by Surface. It is often more convenient 
to locate images, determine focal lengths, and so on, by tracing paraxial (first-order) 
rays through an optical system surface by surface, especially when precise values are 
required. 

Given an optical system with radii r,, r 2 , r 3 , ..., r» separated by distances t\, t' 2 , t ' 3 , 
..., t'„- 1 and an object of height h an axial distance l\ from the vertex of ri, then rays may 
be traced through the system by repeated application of the following equations: 

( u'irii ) = Ui-irii + yiin'i — n,)/r, (9-27) 

yu 1 = yi - t'i(u'i n'i)/ n'i (9-28) 


where u = iz ! + 1 is the slope of the ray after refraction at surface i and y< is the height 
that the ray strikes surface i. 

Focal lengths, focal points, image positions, and image sizes may be found by the 

process described in Sec. 9.2.3.1 and given by Eqs. (9-14) through (9-19). 

Where the axial intercept of a ray is the only item of interest, the following equations 

may be used: , , , 

n n [n — n) 

7“7 + r (9 ‘ 29) 

and 

li + 1 = l'i ~ t'i (9-30) 


More detailed information on ray tracing is provided in Sec. 9.5. 

9.3. Limitation of Rays by Stops and Apertures 

Apertures, or stops, limit the passage of energy through an optical system. 

9.3.1. The Aperture Stop. By following the path of the axial rays in Fig. 9-5, the 
aperture stop can be determined. Diaphragm #1 is the aperture stop of the system. 
This limits the size of the axial cone of energy from the object since all the other ele¬ 
ments are large enough to accept a bigger cone. The ray through the center of the 
aperture stop is called the principal or chief ray and is used to locate the pupils. The 
entrance and exit pupils of the system are the images of the aperture stop formed by 
all elements in object and image space, respectively. In Fig. 9-5, the entrance pupil 
lies in the objective lens and the exit pupil to the right of the eye lens. The inter¬ 
section of the principal ray with the axis locates the pupils, and the diameter of the 
axial cone at the pupil indicates the pupil diameter. 



Wall or 
Structure 


Diaphragm #1 


Erector 


Diaphragm #2 


Entrance 

Window 


Principal Ray 


Object 


Entrance 

Pupil 


/ 


Objective 


Eye Lens 


Exit Pupil 


Fig. 9-5. Optical system demonstration entrance and exit 
pupils and windows, field stop, and glare stops. 



















378 


OPTICS 


9.3.2. The Field Stop. By tracing the path of the principal ray in Fig. 9-5, the 
field stop can be determined. Diaphragm #2 is the field stop since this diaphragm 
would prevent a principal ray starting from a point in the object that is further from 
the axis from passing through the system. The images of the field stop in object and 
image space are the entrance and exit windows, respectively. In Fig. 9-5, the entrance 
window is coincident with the object and the exit window is at infinity (which is coinci¬ 
dent with the image). The windows of a system do not always coincide with object 
and image. 

The angular field of view of an optical system is the angle that the entrance or exit 
window subtends from the center of the entrance or exit pupil, respectively. 

9.3.3. Vignetting. In Fig. 9-5 the roles played by the various elements of the sys¬ 
tem are definite and clear cut. In a real system, the diaphragms and lens apertures 
often play dual roles. 

In the system shown in Fig. 9-6(a), the situation is clear for the axial bundle of rays: 
the aperture stop is the clear aperture of lens A, the entrance pupil is at A and the 
exit pupil is the image of A formed by lens B. Some distance off axis, however, the 
cone of energy accepted from point D is limited on its lower edge by the lower rim of 
lens A and on its upper edge by the upper rim of lens B. This effect is called vignetting, 
and the appearance of the system aperture when viewed from the object is shown in 
Fig. 9-6(6). For some object point still farther from the axis, no energy would pass 
through the system. 


Image of Lens B 



(b) 

Fig. 9-6. Vignetting, (a) The limitation of the rays from an off-axis 
point D by the aperture of components A and B. (b) Appearance of 
the system aperture when viewed from point D. 









LIMITATION OF RAYS BY STOPS AND APERTURES 


379 


Thus, for off-axis points, the entrance pupil has become the common area of two 
circles, one the clear diameter of lens A and the other the diameter of B as imaged by 
lens A. The dashed lines in the figure indicate the location and size of the image of 
B, and the arrows indicate the effective entrance pupil, which has a size and position 
completely different than that for the axial case. 

9.3.4. Glare Stops and Baffling. A glare stop is an auxiliary diaphragm located 
at an image of the aperture stop to block out stray radiation. In Fig. 9-5 the objective 
diameter could be reduced sufficiently so that it is the aperture stop. Reflected radia¬ 
tion is blocked out by diaphragm #1, acting as a glare stop, if it is an accurate image 
of the objective, since the stray radiation will appear to be coming from a point outside 
the objective aperture. Another glare stop could be placed at the exit pupil in this 
particular system. 

In an analogous manner, an auxiliary field stop could be placed at image #1 to 
further reduce stray radiation. 

Baffles are often used to reduce the amount of stray radiation in a system. In Fig. 9-7, 
if radiation from outside the field of view is a problem, the walls of the system can be 
baffled as shown in the lower part of the figure. The key to efficient use of baffles is 
to arrange them so that no part of the detector can "see” a surface which is directly 
illuminated. The short-dash lines of the figure indicate the stray radiation from the 
lens. The long-dash lines indicate the limits of the portion of the system visible to 
the detector, and the solid lines indicate the limits of the illumination which can pass 
through the lens. 


_Ray Paths of Radiation Illuminating the Inside of the Housing 

- - - Baffles Cutting Off Illuminated Areas from Direct View of Detector 



Fig. 9-7. Infrared detector system, showing baffles reducing stray light. 


9.3.5. Relative Aperture, Speed, F/No, Numerical Aperture. As noted in Sec. 
9.3.1, the irradiance at the image formed by an optical system is limited by the size 
of the aperture stop. It is in fact determined by the relative size of the aperture stop, 
and thus the illuminating power of an optical system is frequently expressed in terms 
of relative aperture. 

For a system with its object at infinity, the relative aperture is given by the ratio of the 
effective focal length to the clear aperture (diameter of the entrance pupil). Thus 

f !no = e.f.l./clear aperture (9-31) 












380 


OPTICS 


Another way of expressing this relationship is by the numerical aperture, which is 
the index of refraction (of the medium in which the image lies) times the sine of the 
half-angle of the cone of illumination. Thus 

numerical aperture = N.A. = n sin U (9-32) 

For aplanatic systems, one free from spherical aberration and coma, numerical aperture 
and ft no are related by 1 

/'/ no =2N A (9_33) 

The irradiance on the image is directly proportional to the reciprocal of the square 
of the ft no, and is given by (see Fig. 9-8) 

H = tttN sin 2 U = 0-34) 

4(//no) 2 

where H is the irradiance at the image 

r is the transmission of the system 
N is the radiance of the object 



Fig. 9-8. Relationships between ft no 
= e.f.l./clear aperture — 1/2 N.A. = 1/2 
n sin u . 

9.3.6. Depth of Field and Depth of Focus. The concept of depth of field assumes 
that for a given system there exists a blur small enough that it will not affect the system 
performance. It is then of interest to determine the amount of defocusing which cor¬ 
responds to this blur size and which can thus be tolerated. In photography the blur 
size is conventionally expressed in terms of its linear dimension. In infrared work 
the concept of an angular blur size is more useful. Thus, if a system with clear aperture 
A will tolerate an angular blur of /3 radians, one can see from Fig. 9-9 that 

8 = (3D 2 /A (9-35) 

where 8 is the distance the object can be shifted from its focused position before it 
introduces an angular blur of /3, and D is the distance from aperture A to the object. 
(D is assumed large compared to 8.) 



System 

Aperture 


Fig. 9-9. Depth of focus S resulting from a tolerable angular blur. 

























ABERRATIONS 


381 


Similarly, 

8' = (3D' 2 /A = (if 2 /A = (3f(f/ no) (9-36) 

which is applicable when the object is at infinity and the distance ( D ') from aperture 
to image is equal to the focal length. 

If 8 is not small compared with D, then 


8 = pD 2 /(A ± (SD) 


(9-37) 


and the depth of focus toward the optical system will be smaller than that away from 
the system. 

9.4. Aberrations 

Aberrations are departures from perfect imagery in an optical system. A perfect opti¬ 
cal system would bring all the rays from a point object to focus at the gaussian image 
point; because of aberrations some of the rays do not focus at the proper image point. 

9.4.1. Aberration Description. An aberration is often expressed as a longitudinal, 
transverse, or angular aberration. Figure 9-10 uses spherical aberration to show the 
relationship among the three measures of that aberration. 

See Fig. 9-10 for the relationship expressed as follows: 


r „ TA AA-f 

L //1 — — 

tan U'm tan U' m 


(9-38) 


where f is the system focal length (or the distance from the second principal plane to 
the focal plane if the system is working at finite conjugates), and LA, TA, and A A 
refer to longitudinal, transverse, and angular aberrations, respectively. 



Fig. 9-10. Relationship between longitudinal, transverse, and 
angular aberration (uncorrected spherical aberration). 


Aberrations contributed to a system by a particular element, component,or member 
are transferred through the system according to the laws of magnification, as outlined 
in Sec. 9.2.1.2. The angular aberration at the final image is the sum of the angular 
aberrations of all the contributing elements of the system. 

Aberrations are often described as undercorrected or overcorrected. An aberration 
which is similar in direction or sign to that of a simple positive lens is usually called 
undercorrected. The sign convention for each aberration is given with the descrip¬ 
tions of the individual aberrations in Sec. 9.4.2. 














382 


OPTICS 


9.4.2. The Seidel Aberrations. The seven basic aberrations defined below are 
usually referred to as the Seidel or the primary aberrations. 

Mathematically, transverse aberrations can be expressed as a series expansion in 
terms of the angle U' m between the marginal ray and the optical axis. The first-order 
term corresponds to a focusing effect and can be eliminated by a shift of the image 
reference plane (usually to the position of the paraxial focus). The remaining terms 
of third, fifth, seventh, and so on, powers of U' m form a rapidly converging series of 
which the third-order term is very often predominant. This third-order term is thus 
a useful approximation of the image aberrations of a system, and the third-order aberra¬ 
tions, as they are called, can be calculated from data derived from a paraxial ray trace. 

9.4.2.1. Spherical Aberration. Spherical aberration is the variation of focus with 
aperture, in which a ray through the margin of the lens intersects the axis at a point 
other than the paraxial focus. A system is undercorrected if the marginal ray comes to 
a focus before the paraxial focus (see Fig. 9-10). The sign convention for spherical 
aberration is given by 

LA'=L'-l' (9-39) 

where L' is the axial intercept of the marginal ray and /' is the paraxial intercept. The 
image formed by a system with spherical aberration is a circular blur. 

9.4.2.2. Coma. Coma is the variation of magnification (or focal length) with 
aperture. Because of coma, rays passing through the margins of the lens intersect the 
final image plane at a different height than the principal ray (which passes through 
the center of the aperture). The upper and lower rim rays of a comatic system inter¬ 
secting the image plane below the principal ray and the appearance of a typical coma 
patch are shown in Fig. 9-11. The size of the coma patch is given by 

coma, = H'ab ~ H'pr (9-40) 

The correspondence between the position of a ray as it passes through the aperture of 
a system and its position in the coma patch is shown in Fig. 9-12. As the ray position 
moves 90° from A to C in the aperture, its position moves 180° in the coma patch. The 
distance from AE to PR in the coma patch is called tangential coma (coma,) and the 
distance from CG to PR is called sagittal coma (coma,). For third-order coma 

coma, = 3 • coma, (9-41) 


In a typical coma patch 50 to 60 percent of the energy is contained in the pointed end 
of the patch, between PR and CG. 

The Abbe sine condition states that, for an optical system to have perfect imagery 
in the region near the optical axis (but not at the axis), the ratio sin U x : sin U' k must 
be constant for all rays. The offense against the sine condition (O.S.C.) is a convenient 
measure of coma in the region of the optical axis and is given by 


O.S.C. 


coma, _ sin U x u k 
h U\ sin U'k 


l' ~ l' P r 
(L' - Vpr) 


(9-42) 


where U x and U\ are the slope angles of the marginal and paraxial rays from the axial 
object point, U' k and u' k are the corresponding angles in image space, L' and 1 are the 
axial intercepts of the marginal and paraxial rays, and l' pr is the final intercept of 
the principal ray (that is, the position of the exit pupil). The quantity (sin U x )/u x 
is equal to Yly (the ratio of the ray heights) when the object is at infinity; also, if the 
exit pupil is at the last surface of the system, l' pr = 0. 






ABERRATIONS 


383 




H' 

pr 


Fig. 9-11. Upper and lower rim rays of comatic system intersecting the 
image plane below principal ray (a); coma patch (6). 


A 



(b) 




























384 


OPTICS 


9.4.2.3. Astigmatism and Field Curvature. Astigmatism in an off-axis image of 
a point is the difference in focus between the fan of rays in the plane of the axis and 
the object point (the meridional or tangential fan) and the fan of rays perpendicular 
to this plane (the sagittal or skew fan). In Fig. 9-12(a), rays in the plane AA'-E'E 
of the aperture constitute the tangential fan, and rays in the plane CC'-G'G constitute 
the sagittal fan. Astigmatism is undercorrected when the tangential fan is brought 
to a focus before the sagittal fan. The appearance of the astigmatic image is a line at 
either focus: at the sagittal focus the image is a radial line, which, if extended, would 
pass through the axis; at the tangential focus the image is a tangential line perpendic¬ 
ular to the radial line. Between the foci the image is diamond-shaped. 

The curvature of field is the distance parallel to the axis from the focus of an off-axis 
image to the axial focal plane. The sagittal curvature of field is given by 

X s = L'cg ~ l' (9-43) 

and the tangential curvature by 

X, = L'ae ~ I' (9-44) 

where X is the departure from the plane of the surface of the focus 

L' is the axial distance from the last surface of the system to the axial projection 
of the intersection of the subscript rays 

/' is the distance to the paraxial focus 
s, CG, t, and AE refer to Fig. 9-12. 

An optical system composed of a set of elements of given power and index has a basic 
field curvature called the Petzval curvature. Although the astigmatism of a system 
can be changed by bending the elements, the Petzval curvature cannot be changed to 
any great extent. For regions near the axis, the tangential focus is always three times 
as far from the Petzval surface as the sagittal focus, satisfying the relationship 

X t - Xptz = 3(X, - Xptz) (9-45) 

where X p tz is the curvature of the Petzval surface. 

9.4.2.4. Distortion. Distortion is the departure of the image height from that 
predicted by first-order gaussian optics. The image of a rectangular figure takes on 
the shape of a pillow or pincushion with concave sides in the presence of distortion that 
causes an enlargement of the image. Distortion of the opposite sign produces an image 
with convex sides, like a barrel. Distortion is given bj' 

distortion = H' pr — h' (9-46) 

where H' pr is the intersection of the principal ray with the paraxial image plane 
h' is the paraxial image height. 

9.4.2.5. Axial (Longitudinal ) Chromatic Aberration. The images formed by an 
optical system may have different sizes and positions for different wavelengths. Be¬ 
cause the index of refraction of optical materials varies with wavelength, longitudinal 
chromatic aberration is the difference in focal position between images formed by 
two different wavelengths, and is given by 

LchA = l' v - V r (9-47) 

where l' v is the final image distance for the shorter wavelength 
l'r the image distance for the longer wavelength. 


ABERRATIONS 385 

9.4.2.6. Off-Axis ( Transverse ) Chromatic Aberration. Off-axis aberration results 
in a difference in image size due to wavelength variation and is given by 

TchA = H' v — H' r (9-48) 

where H' v — paraxial image plane intersection of principal rays of short wavelengths 
H'r — paraxial image plane intersection of principal rays of long wavelengths 

Transverse chromatic, or lateral color can be considered as a chromatic difference 
of magnification (C.D.M.) and expressed as: 

C.D.M. = (H' v - H' r )/ K (9-49) 

9.4.3. Correction of Aberrations. In practical optical systems, aberrations are 
usually corrected by balancing the undercorrection of one element against the overcor¬ 
rection of another. Because an optical system should have a given power of focal length, 
it is necessary to combine elements whose Xy4> (see Sec. 9.2.3.1) equals the desired 
power, but whose summed aberrations equal zero. In chromatic aberration in a 
doublet, a positive element with a certain chromatic aberration contribution per unit 
of power is combined with a negative element with a relatively higher chromatic con¬ 
tribution per unit of power, so that the chromatic contributions are equal and opposite 
and cancel while leaving a residue of power. 

Chromatic aberrations are corrected by proper choice of materials, element powers, 
and spacings. The Petzval curvature is corrected by choice of material and element 
powers. The preceding aberrations are ordinarily corrected in the designer’s initial 
layout of the powers and spacings of the optical elements to be used in the system, at 
the same time that the system power and working distance are arranged. Spherical 
aberration, coma, astigmatism, and distortion can be controlled by proper shaping or 
bending of the elements of the system. Aspheric surfaces (that is, surfaces of revolution 
which are not spheres) may also be used to correct the monochromatic aberrations. 

9.4.4. Variations of Aberrations with Aperture and Image Size. The amount of 
a primary aberration in the image is a function of the semiaperture (y) of the system 
and the height (h ) of the image. Table 9-1, based on third-order aberrations, is useful 
in estimating the effect of a change in aperture or field coverage (image size) on the 
performance of a system. 

Table 9-1. Variation of Aberrations with Size of Aperture, Field Angle, 
and Image Size, for System with Pure Third-Order Aberrations 



Size of 

Field 

Image 


Aperture 

Angle 

Size 

Aberration 

(y) 

(u) 

(h) 

Longitudinal spherical aberration 

y 2 

— 

— 

Transverse spherical aberration 

y 3 

— 

— 

Coma 

J' 2 

u 

h 

Astigmatism 

— 

u 2 

h 2 

Length of astigmatic focal lines 


u 2 

h 2 

Petzval curvature 

— 

u 2 

h 2 

Distortion 

— 

u 3 

h 3 

Percentage distortion 

— 

u 2 

h 2 

Longitudinal axial chromatic aberration 

— 

— 

— 

Transverse axial chromatic aberration 


— 

— 

Lateral chromatic 

— 

u 

h 

Chromatic difference of magnification (C.D.M.) 

— 

— 

— 


386 


OPTICS 


9.4.5. Zonal and Residual Aberrations. When an aberration is fully corrected 
for a certain region of the aperture or portion of the field, there usually remain aberra¬ 
tions for rays passing through other parts of the aperture or for smaller or larger 
field angles. 

9.4.6. Chromatic Variation of Aberrations. Because the index of optical materials 
changes with wavelength, the monochromatic aberrations (spherical, coma, astigma¬ 
tism, Petzval curvature, and distortion) will also vary with the wavelength of radiation 
passing through an optical system. Chromatic variation of spherical aberration 
(spherochromatism) is most common. Ordinary spherochromatism causes the spherical 
aberration in the shorter wavelength to be more overcorrected than that of the longer 
wavelengths. 

9.4.7. Graphical Representation of Aberrations. Spherical and chromatic 
aberration can be presented as longitudinal or transverse aberrations, the former 
being plotted against the entering ray height (or angle) and the latter against the 
slope angle of the emergent ray. The spherical aberration of a single element with 
both types of graphical presentation is shown in Fig. 9-13. 


Plot of Longitudinal 
Aberration (LA 1 ) vs. 



Fig. 9-13. Undercorrected spherical aberration. The longitudinal spherical aberration is 
plotted against the entering ray height Y. The transverse spherical is plotted against 
the tangent of U', the final angle the light ray makes with the axis. 


The plot of H' vs. tan U' is called a rim ray curve and is useful because it indicates 
directly the size of the blur in the image caused by the aberration; and the effect of 
refocusing (or shifting the reference plane) can be readily determined by rotating the 
X or tan U' axis. 

Curvature of field is ordinarily represented by plotting the longitudinal difference 
of focus between axial and oblique rays against either the image height or the field 
angle, as shown in Fig. 9-14. 

The rim ray curve can also be used to represent the aberrations of oblique fans of 
meridional rays, as shown in Fig. 9-15. The shape of the rim ray curve is indicative 
of the aberration present, as shown in Fig. 9-16 for common aberrations. 

Meridional aberrations can be expressed in terms of X and Y. Skew or sagittal 
fans have aberrations in the Z direction also, and the representation of the aberrations 
of skew rays is more complex. Two methods are common. One is to plot the Y-Z 


























ABERRATIONS 


387 



Fig. 9-14. Relationships between sagittal and tangential focal surfaces 
and Petzval surface for lens with undercorrected astigmatism. 



Fig. 9-15. Construction of off-axis rim ray curve to represent aberrations 
of oblique fans of meridional rays. 




Fig. 9-16. Appearance of rim ray curve in presence of 
typical aberrations: (a) spherical aberration, third-order 
undercorrected, (6) spherical aberration, third-order over¬ 
corrected, (c) zonal spherical aberration, (d ) third-order coma, 
(e) defocusing or field curvature, (/") rim ray curve for a 
typical oblique fan showing coma, field curvature, and oblique 
spherical. 


(e) 


(f) 
















388 


OPTICS 


coordinates of the ray intersections with the image plane, connecting the points made 
by rays which lie along a radial line in the aperture. If a large number of points is 
plotted, the density of points is representative of the flux density in the image, and 
the flux density in the image and the spot diagram become a "picture” of the image 
(provided that diffraction can be neglected). In another method, the Y and Z departures 
are plotted separately against the radial distance from ray to the center of the aperture. 
The rim ray curve described above is this type of "clock plot” for the meridional fan. 
A clock plot is useful in aberration analysis and in interpolating for spot diagrams. 

9.5. Ray Tracing 

Ray tracing is used to calculate the path of a ray of light through an optical system. 
It is ordinarily done either to evaluate an optical system or as a step in the process of 
optical design. 

Ray tracing is based on Snell’s law (Eq. 9-1) and can be done by using geometry to 
calculate the ray paths between surfaces and applying Snell’s law at each surface. 
Specific equations for this purpose are given in Sec. 9.5.3. A crude form of ray tracing 
may be carried out with drawing tools; for the technique see Sec. 9.5.5. 

9.5.1. Ray-Tracing Precision. Depending upon the scale of the calculation, the 
precision required of dimensional numbers is to four, five, or six decimal places. Angles 
and trigonometrical functions should have six-figure accuracy. Trigonometric tables 
given in terms of radians are preferred for ray tracing with a desk calculator. Most 
trigonometric ray-tracing results are referred to the paraxial focus or focal plane. 
The paraxial ray trace is carried out with equations derived from the trigonometric 
ray-tracing equations by setting the sine and tangent equal to the angle, and cosines 
equal to unity. 

9.5.2. Determination of Specific Aberrations. The following subsections outline 
the methods of obtaining numerical values for the basic aberrations. 

9.5.2.1. Spherical Aberration. A paraxial ray and a marginal ray (a trigonometric 
ray through the rim of the entrance pupil) are traced starting at the axial point of the 
object. The final axial intercepts of the rays are to be determined. Longitudinal spher¬ 
ical aberration is given by 

LA' = L' - l' (9-50) 

Transverse spherical aberration is the height at which the ray strikes the paraxial 
image plane and can be found from 

TA' = LA' tan U' (9-51) 

Zonal spherical aberration is found by tracing a ray at a lesser height than the marginal 
ray (usually at 0.707 of the marginal height) and substituting the final zonal-ray data 
in Eqs. (9-50) and (9-51) in place of the marginal-ray data. 

9.5.2.2. Coma. Tangential coma is evaluated by tracing three rays of a meridional 
fan from an off-axis object point through the system. The principal ray ( pr ) passes 
through the center of the entrance pupil, and the upper and lower rim rays (A and B) 
are traced through the upper and lower edges of the pupil. The intersection of ray A 
with ray B is determined, and the distance from the intersection to the axis ( H' AB ) 
is compared to the height of the intersection of the principal ray with a plane through 
the AB intersection ( H ' pr ). The axial distance from the last surface to the AB inter¬ 
section can be found from 

, _ L'a tan U' A — L' b tan U' B 

tv AB — 


tan U'a — tan U' B 


(9-52) 



RAY TRACING 389 

and the intersection heights from 

H'ab = (L' a - L' ab ) tan U'a (9-53) 

H pr ( L pr L ab) tan U p r (9-54) 

Then tangential coma is given by 

comar = H'ab ~ H' pr (9-55) 

The offense against the sine condition (O.S.C.) can be determined from the same ray 
trace used to find spherical aberration: 


O.S.C. 


sin U i u' (/' — I'pr) 
U\ sin U' (L' — V pr ) 


For regions close to the optical axis, 


coma, = 3 coma., = 3H' O.S.C. 


(9-56) 


(9-57) 


9.5.2.3. Astigmatism and Field Curvature. Curvature of field for small apertures 
is found by tracing the equivalent of a paraxial ray near a principal ray instead of 
near the optical axis. The calculation is carried out using Coddington’s equations 
(see Sec. 9.5.3). 

The extended Petzval surface may be found from the x 8 and x t resulting from a 
Coddington’s trace by 

x ptz = 1.5 jc s — 0.5x, (9-58) 

9.5.2.4. Distortion. Distortion is found by tracing a trigonometric principal ray 
and comparing the height of its intersection with the paraxial focal plane to the image 
height predicted by gaussian optics. The height for the principal ray is found from 


H'pr = ( L'pr - 1') tan U' pr (9-59) 

For objects at infinity the gaussian height is given by 

h' = -f tan U pr (9-60) 

and for an object at a finite distance by 

h' = hnuln'u' (9-61) 


Distortion is then 


dist = H'pr — h' 


(9-62) 


Distortion is often specified as a percentage of the image height. 

9.5.2.5. Longitudinal Chromatic Aberration. This aberration is found by locating 
the image position for the different wavelengths of interest by using the appropriate 
values for the index of refraction in the calculation. The calculation may be paraxial 
or trigonometric. For paraxial rays 

LchA = l'v-1'r (9-63) 


where v and r refer to the short and long wavelengths, respectively. If marginal 
rays are traced in various wavelengths, it is useful to make a plot of L' vs. Y, or H' 
vs. tan U' (see Sec. 9.4.7) for all wavelengths on the same graph, because this will 
also indicate the spherochromatism. 





390 


OPTICS 


9.5.2.6. Lateral Color, or Transverse Chromatic Aberration. Lateral color is found 
by tracing principal rays for the long and short wavelengths. The lateral color is the 
difference between the paraxial focal-plane intersection heights of the two principal 
rays and is given by 


TchA = H' v — H' r (9-64) 

where H' v and H', are determined from Eq. (9-59). 

9.5.3. Ray-Tracing Equations. 

9.5.3.I. For Desk Calculators. Paraxial ray-tracing equations (see Fig. 9-17): 
Opening: 

n x u\ = niyJh (9-65) 

Iterative: 

n'iu'i = niUi + yiin'i — nj)/rj (9-66) 

y.+i = y> — t'm' jU 'i/n'i (9-67) 

Closing: 

l'i = yin'il n'iu'i (9-68) 

If U\ = 0, then 

e.f.\.=yin'k/n'ku'k (9-69) 

b.f.l. = y k n' k ln ' k u' k (9-70) 



Fig. 9-17. Quantities used in paraxial ray tracing. 


For speed, the quantity nu is carried as an entity. If the values of the angles of in¬ 
cidence and refraction are required for third-order aberration calculations, they can 
be found from 


i = y/i — u 


(9-71) 


i' = niln' (9-72) 

Meridional ray-tracing equations for finite radii (see Fig. 9-18): 

Opening: — 

CA = (L — R) sin u 



(9-73) 











RAY TRACING 


391 


Normal to 
Surface 



Iterative: 


Closing: 


sin I, = CAi/rt 


T, Hi 

sin I i = — sin I j 

n i 

U i+ 1 = U'i 


CAj+i = (r* — r,+i — £',) sin U'i + 


riiCAi 


n j 


L\ = n + — 


n jCA i 


n'i sin U'i 


Miscellaneous: 

Coordinates of ray intersection with surface: 

Y = r sin (U + I) 

X = 2r sin 2 \ (U + I) 


Distance between surfaces along ray: 

Di to i+l = (t id” ^i + l Xi) Sec U i 

or 

Di to i+i = n cos I'i — r, + i cos Ii+i — (r, — r,+i — t'i) cos U'i 


(9-74) 

(9-75) 

(9-76) 

(9-77) 

(9-78) 


(9-79) 

(9-80) 

(9-81) 

(9-82) 





















392 


OPTICS 


Plane 



Fig. 9-19. Quantities used in meridional ray-trace equations 
for plane surfaces. 


Meridional ray-tracing equations for plane surfaces (see Fig. 9-19): 
Opening: 

OPi = L\ sin U i 


or if t/x = 0, OPt = Yt. 

Transfer from preceding radius: 


Iterative: 


Closing: 


n i — 1 i — 1 i / .f \ • T TI 

OPi =---1- (n -1 — t'i-i) sin U'i-i 

n i-i 

Ui= 


sin U'i = — sin Ui 

n'i 


OP' i = 
OP i+1 = 

CAi+i = 


cos U'i 


OPi 


cos Ui 
OP'i — t'i sin U'i 
OP'i + (—rj+i — t'i ) sin U' t 


L'i = OP'J sin U'i 


Miscellaneous: 


(9-83) 


(9-84) 

(9-85) 

(9-86) 

(9-87) 

(9-88) 

(9-89) 

(9-90) 


Yt = OPi/ cos Ui (9-91) 

Di to j+i = r< cos I'i - OP i+ 1 tan U'i — (r< — t'i ) cos U't (9-92) 

Ditoi+i = ~ (— ri+i — t'i) cos U'i + OP'i tan U'i — r i+i cos 7<+i (9-93) 

Coddington's equations (for tracing close sagittal and tangential rays about a 
principal ray for determination of astigmatism and field curvature) (see Fig. 9-20): 
A principal ray is traced using Eqs. (9-78) through (9-93) (as applicable). 
















RAY TRACING 


393 



The "oblique power,” <f>, of each surface is calculated from 

<f> = (N' cos I'p — N cos I p )/r (9-94) 

The sagittal rays are traced by repeated application of 

n' n , . 

7 = 7 + * (9-95) 

and 

Si + i = S i Di to i+ 1 (9-96) 

The tangential rays are traced by repeated application of 

' *'eo*r, = ncos*I p +<j> (9 . 97) 

and 

t i+1 = t'i- Di to i+i (9-98) 

where s and t are the distances along the principal ray from surface to focus. 

Under the usual sign convention the ray trace is started with Si = t t equal to 
a negative value if the object is to the left of the first surface. 












394 


OPTICS 


For an object at infinity, Si = t\ = infinity. 

For a finite object distance s i = t\ — (L x — X pri ) sec U pri , where X pr is found 
from Eq. (9-80). 

The quantity D is the distance along the ray from surface to surface and is found 
by Eqs. (9-82), (9-92), or (9-93). 

Closing: The final curvature of field is found from 


X' s = s'k cos U' Pk + X Pk — l' k (9-99) 

X't = t' k cos U' Pk + X Pk - l'k (9-100) 

9.5.3.2. For Electronic Computers. The paraxial equations in Sec. 9.5.3.1 are 
suitable for electronic computers. 

Meridional ray-tracing equations (see Fig. 9-21): 

Opening: Q is the perpendicular to the ray from the vertex (axial intersection) of the 
surface; thus 


Q i = L i sin U i 
Iterative: c is the reciprocal of the radius of curvature. 

sin I = Qc — sin U 
cos / = V1 — sin 2 I 
sin (t/+7) = cos U cos I + sin U sin I 
cos ( U+I) — cos U cos I — sin U sin I 
sin I' = ( nln') sin I 
cos /' = Vl — sin 2 /' 

sin U' = sin ( U+I ) cos /' — cos (£/+/) sin /' 

cos U' = cos (U+I) cos /' + cos (U+I) sin I' 

Q, _ Q (cos U' + cos I') 

(cos U + cos I) 

Qi+i = Q'i — t'i sin U'i 


(9-101) 

(9-102) 

(9-103) 

(9-104) 

(9-105) 

(9-106) 

(9-107) 

(9-108) 

(9-109) 

(9-110) 

(9-111) 



Fig. 9-21. Quantities used in computing formulas for electronic computers. 

















RAY TRACING 


395 


The intersection height of the ray with the surface may be found from 


Y -q> [1 + cos (U+I)] 


(9-112) 


Closing: 


(cos XJ' + cos /') 

L'i = QVsin U\ (9-113) 

To find the height the ray strikes a plane a distance 7' from the surface i, use 


H' = (Q'i~ sin U’d/coa U’t (9-114) 

Coddington’s equations (close skew and close meridional rays): For electronic 
computers Coddington’s equations have been rewritten in a form which does not 
contain the quantities s or i directly. A principal meridional ray is traced by the 
equations of Sec. 9.5.3.2. concurrently with the Coddington trace; the ordinary ray¬ 
tracing quantities used in the following refer to data of this principal ray. 

Opening: 

p s = n t y, ) (9-115) 

Pt = ruy, cos 2 h (9-116) 


where y is an arbitrarily chosen ray height from the principal ray, analogous to 
the y of the paraxial ray trace, and Si and t\ are the distances from the object to 
the first surface along the principal ray, and P s and P t are the tangential and sagittal 
representations for the principal ray. 

Iterative: 


4>i = Ciin'i cos l'i — ni cos /,) 
P si — P s(i-l) “I - ysi<f>i 
P'ti =P't(i- 1 ) + yti<fri 
Qi sin (Ui + h) 

Xj = - 

(cos Ui + cos /,) 


Di to i+1 — 


t j to i + l X j -t-1_ X j 

cos U'i 


y*(f+l) — ysi P siDi to i+i/Uj 

— cos2 Pi (yti ~ P 'tiDi to ^ 

+ cos 2 /(i+i) n'i cos 2 l'i 


(9-117) 

(9-118) 

(9-119) 

(9-120) 

(9-121) 

(9-122) 

(9-123) 


Closing: 

s'k = n' k-ysk/P'sk (9-124) 

t'k = n' k ytk cos 2 1'k/P'ik (9-125) 

The curvature of field from a surface a distance l' from the final surface may be 
found from 

X'g = s'k cos U'k + Xk-l' (9-126) 

X' t = t' k cos U'k + Xk-P (9-127) 







396 


OPTICS 


y axis 




Fig. 9-22. Direction cosines and surfaces intercept coordinates (a) 
and subscript convention (6) used in skew ray-trace formulas. 


Skew ray trace through a spherical surface (Fig. 9-22a): The general ray is defined 
by its direction cosines X,Y,Z and the coordinates of its intersection with the surface 
x, y, z. The calculation is usually started with a dummy surface (c = 0 and n= n') 
placed at the entrance pupil for convenience; thus the opening data are the ray 
vectors X, Y, Z and the ray coordinates in this surface. The calculation is closed 
by calculating the ray coordinates x, y, z in a final reference surface which is usually 
the final image plane (or surface). L is the distance along the ray from surface i 
to surface (i+ 1). 

The following equations are applied surface by surface: 


e = tX- xX-yY-zZ 

(9-128) 

M s = x + eX — t 

(9-129) 

M\ 2 = x 2 4- y 2 + z 2 — e 2 + t 2 = 2 tx 

(9-130) 

Ex = Vx 2 — C\{c\M\ 2 — 2 M t ) 

(9-131) 

L = e + (ciMi 2 - 2 M x )l(X + Ex) 

(9-132) 

Xi = x + LX — t 

(9-133) 
















RAY TRACING 

397 

yt = y + LY 

(9-134) 

Z\ — z + LZ 

(9-135) 

Mi = nln 1 

(9-136) 

E ', = Vl - mi 2 (1 -^i 2 ) 

(9-137) 

^1 — E' 1 — p x E 1 

(9-138) 

X\ = giX — g 1 C 1 X 1 + g 1 

(9-139) 

Y x = g x Y - gic x y x 

(9-140) 

Z, = /x iZ — g x c x z x 

(9-141) 


The ray "height” at the surface may be found from 

s ( = Vy, 2 + z i 2 (9-142) 

Unsubscripted quantities refer to surface i and those with the subscript 1 refer to 
surface (i 4- 1). The axial spacing Z is from i to (i+ 1). n is n, + J and n x is n'i +x . E x 
and E'i are the cosines of the angles of incidence and refraction. See Fig. 9-22(b). 

Skew ray trace through an aspheric surface: As above, the general ra}' is defined 
by its direction cosines X, Y, Z and its intersection coordinates x, y, z. The difficulty 
in tracing through an aspheric lies in determining the intersection of the ray and 
the aspheric surface. This is accomplished by successive approximations, the ap¬ 
proximation process continuing until the residual error is negligible. 

The aspheric surface is represented by the expression 

c s 2 

X = f(y, z) =- h A 2 s 2 + A 4 s 4 + . . . (9-143) 

1 + Vl - C 2 S 2 

where c = l/R, s 2 = y 2 + z 2 . The first term is the equation for a spherical surface, 
and A 2 , A. t , A 6 , and so on, are the aspheric deformation constants of the surface. 

The first step is to compute x Q , y«, and z 0 , the intersection coordinates of the 
ray with the sphere (of curvature c) which approximates the aspheric. This can 
be done through use of Eqs. (9-128) through (9-135). Then one calculates 


x 0 = f(yo,z 0 ) (9-144) 

by stubstituting So 2 = yo 2 + Zo 2 into Eq. (9-143). Then one computes 

Zo = (l - c 2 s 0 2 ) 1/2 (9-145) 

m 0 = — yo [c 4- Zo (2 A 2 + 4A 4 S 0 2 + ...)] (9-146) 

no — ~Zo [c + Zo ( 2 A 2 + 4A 4 So 2 +...)] (9-147) 

Go = Zo (x 0 — x 0 )/(Xlo + Ym 0 + Zno) (9-148) 

Xl = G 0 X+Xo (9-149) 

y, = GoY + y 0 (9-150) 

z, = G 0 Z + zo (9-151) 






398 


OPTICS 


The computations of Eqs. (9-144) through (9-151) are repeated, with subscripts 
increased by one each time until 


Xk = x k (9-152) 

to within the accuracy required. 

The refraction at the surface is calculated by 

O 2 = l k 2 + m k 2 + n k 2 (9-153) 

E i = XI k + Ym k + Zn k (9-154) 

E'i = [0 2 (1 - mi 2 ) + Mi 2 ^i 2 1 1/2 (9-155) 

gi = (E'i ~ fMiEi)/0 2 (9-156) 

Xt^/nX+gtl (9-157) 

Yi = iixY + g x m k (9-158) 

Z l = f i l Z+g i n x (9-159) 


Then the direction cosines Xi, Y u Z x and the intersection coordinates x k , y k , z k 
define the refracted ray. 

9.5.4. Graphical Ray Tracing. This technique applies Snell’s law (n sin /= n' sin I ') 
at each surface of the system. It may be carried out by drawing a ray to its intersection 
with the surface, constructing the normal to the surface, measuring the angle 1 with 
a protractor, and then calculating the angle /'. 

A purely graphic technique is shown in Fig. 9-23. The ray is-drawn to the surface 
and the normal to the surface is erected at the point of intersection. Two circles are 
drawn about the point of intersection with radii proportional to the indices on either 
side of the surface. From the intersection of the ray with circle n at A , a line is drawn 
parallel to the normal to intersect circle n ' at B ; then the refracted ray is drawn through 
B and the ray-surface intersection. 



Fig. 9-23. Graphical ray tracing. Starting with the construc¬ 
tion of circles (with radii proportional to the indices on either 
side of the surface) about the point of intersection of the ray and 
surface and the development of the refracted ray. 



THIRD-ORDER ABERRATIONS 


399 



9.6. Third-Order Aberrations 

Figure 9-24 illustrates typical third-order aberrations. 

9.6.1. Third- and Higher-Order Aberrations — Surface Contributions. The con¬ 
tribution of a given surface to the third-order aberration of the final image can be calcu¬ 
lated by the equations in the following sections. The final third-order aberration is the 
summation of the contributions of all the surfaces. Two paraxial rays are traced 
through the system; one is the ray from the axial object point passing through the edge 
of the entrance pupil, a paraxial marginal ray, and the other is the ray through the 
center of the entrance pupil from an object point at the edge of the field, a paraxial 
principal ray. In the equations, quantities with the subscript pr denote the data of 
the paraxial principal ray; quantities without subscript refer to the data of the paraxial 
marginal ray. 











400 


OPTICS 


The equations are evaluated for each surface, from the ray-trace data of that surface. 
The contributions are then summed to obtain the total third-order aberration of the 
system. 

9.6.I.I. Equations for Desk Calculator. 

Spherical contribution: 

SC' = yni(i' - i) (f - u)/2n' k u' k 2 (9-160) 

SC' = yu'n'iu 2 - u' 2 )/2n' k u' k 2 (for a plane) (9-161) 

Coma (sagittal) contribution: 

CC'g = SC'Ru'k (9-162) 

Astigmatism (sagittal) contribution: 

AC's = SC'R 2 (9-163) 

= ynui p 2 (n' — n)l n'2n' k u' k 2 (9-164) 

(for use when object is near the center of curvature). 

Petzval contribution: 

PC' = (n — n')h k 2 n' k /2nn' r (9-165) 

Distortion contribution: 

DC' = (AC' + PC')Ru' k (9-166) 

Longitudinal chromatic contribution: 

/An. A n'\ / 

LchC' = yni ( —-—) / n' k u\ 2 (9-167) 

Lateral color contribution: 

TchC' = (LchC')Ru' k (9-168) 

The symbols above are defined as follows: 

R = ip/i 

— Up/u (for a plane) (9-169) 


SC' is the longitudinal third-order spherical aberration contribution. 

CC' is the sagittal third-order coma contribution and is equal to one-third of the 
tangential coma contribution. 

PC' is the contribution to the third-order Petzval curvature and jc p t z = SPC'. 

LchC' is the axial longitudinal chromatic aberration and XLchc' is equal to l' v — l' r . 
TchC' is the transverse olf-axis chromatic contribution and S^chC' = h' v — h' r . 

AC' is the longitudinal sagittal astigmatism. 

The third-order field curvatures may be found from 

x x = XPC' + lAC' 


x, = XPC' + 3 lAC' 


(9-170) 



THIRD-ORDER ABERRATIONS 


401 


r is the radius of curvature, y is the ray height at the surface, n is the index of refrac¬ 
tion, i is the angle of incidence, and u is the ray-slope angle; these are obtained from 
the ray-trace data. Rays may be traced by Eqs. (9-174) through (9-181). The data 
of the final image are primed and subscripted with k. Thus h' k , u'k, and n'k refer to 
the final image height ( i.e ., the height at which the paraxial principal ray strikes the 
final image plane), the final ray-slope angle at the image, and the index of the image 
space, respectively. 

The dispersion of the medium is represented by An = n v — n r , where the subscripts 
refer to the short (u) and long (r) wavelengths of light. For visual work, F and C light 
(0.486 /u, and 0.656 /x, respectively) are customarily used for these wavelengths. 

9.6.I.2. Equations for Electronic Computers or Desk Calculators. 


c = 1/r (9-171) 

N = n/n' (9-172) 

u' = cy(l -N) + Nu (9-173) 

yt +1 = yi — t'iU'i (9-174) 

i = cy — u (9-175) 

h'k = n t (uiy Pl — yiU Pt )ln' k u'k (9-176) 

1= n(uy p — u p y ) = invariant (9-177) 

Z = (N — 1 )c/n (9-178) 

B = ny{u' - i)(l - AO/27 (9-179) 

B p = ny p (u' p - i p )( 1 - N)/2I (9-180) 

TSC = Bi 2 h' k (9-181) 

SC = TSC/u'k (9-182) 

CC = Bii p h'k (9-183) 

TAC = Bi p 2 h' k (9-184) 

AC = TAC/u'k (9-185) 

TPC = ZIh'k/2 (9-186) 

PC = TPCIu'k (9-187) 

DC = h' k [B p ii p + ( u ' p 2 - u p *)/2] (9-188) 

TLC = yi(An-NAn')/u' k (9-189) 

LchC = TLCIu'k (9-190) 

TchC = yi P (An — N An')u' k (9-191) 


The symbols above have the same meanings as in Sec. 9.6.1.1. The contributions 
TSC, TAC, TPC, and TLC are the transverse aberrations for spherical aberration, 
astigmatism, Petzval curvature, and longitudinal chromatic aberrations (which are 
customarily expressed as longitudinal aberrations). The transverse aberration is 
simply the longitudinal aberration times the final ray slope angle, u' k . 


402 optics 


Contributions from an aspheric surface : The aspheric surface 

is defined by 

x = (1/2 )c e s 2 4- [(l/8)c e 3 + if]s 4 -+-... 

(9-192) 

in which the terms in s 6 and higher may be neglected. c e in this expression is not the 
same c as that used in Eq. (9-143) on aspheric ray tracing. For aspherics in the form 
of Eq. (9-143), an "equivalent” c e and the equivalent fourth-order deformation constant 
K can be computed from 

c e — c + 2A 2 

(9-193) 

K = A 4 -A 2 (4A 2 2 + 6 A 2 c + 3c 2 )/4 

(9-194) 

The contributions are determined for the "equivalent” spherical surface by Eqs. (9-171) 
through (9-191). Then the additional contributions due to the "equivalent” fourth- 
order deformation constant K are computed by the following equations and added to 
those of the "equivalent” spherical surface to obtain the total third-order aberration 
contribution of the aspheric. 

W — 4(n — n')K/I 

(9-195) 

TSC a = Wy 4 h' k 

(9-196) 

CC a = Wy 2 y p 2 h' k 

(9-197) 

TACa = Wy 3 y p 2 h'k 

(9-198) 

TPC a = o 

(9-199) 

DCa = Wyy p 3 h' k 

(9-200) 

TLC a = 0 

(9-201) 

Tch.A a = 0 

(9-202) 

9.6.2. Third-Order Aberration Contributions of Thin Lenses. If the thin-lens 
fiction is used {i.e., assuming that the thickness of an element is zero), a useful set 
of aberration-contribution equations for a single element may be derived from the 
preceding equations. The procedure is to trace a paraxial marginal ray and a paraxial 
principal ray through the system using 

u' = u + y<fr 

(9-203) 

yi+ 1 = yi ~ d'iu'i 

(9-204) 

l'k = yk/u'k 

(9-205) 

Then for each element 


v = u/y (or v' = u'ly) 

(9-206) 

Q = y P /y 

(9-207) 

Then the contributions may be determined from 


SC * = SC 

(9-208) 

CC* = CC + SCQu'k 

(9-209) 

AC* = AC + CC2Q/u' k + SCQ 2 

(9-210) 


THIRD-ORDER ABERRATIONS 403 

PC* = PC (9-211) 

DC* = (PC + 3 AC)Qu' k -I- CC-3Q 2 + SCQ*u' k (9-212) 

LchC* = LchC (9-213) 

TchC* = LchCQu'k (9-214) 


The starred terms are the contributions from an element which is not at the stop; that 
is, one for which y p 0. The unstarred terms are the contributions from the element 
when it is located at the stop (and y p = 0) and are given by the following: 

SC = —y 4 (GiC 3 - G 2 C 2 C, + G 3 C 2 v + G 4 CCi 2 - G 5 CC, v + G e Cv 2 )/u ' k 2 

(9-215) 

[or SC = —y 4 (GiC 3 + G 2 C 2 C 2 - G 3 C 2 v' + G 4 CC 2 2 - G,CC 2 v' + G 6 Cv' 2 )/u' k 2 ] 


CC = —h' k y 2 (^G$CCi - G 7 Cv - G 8 C 2 ) 

(9-216) 

[or CC = -h' k y 2 (±G,CC 2 - G 7 Cv' + G 8 C 2 )] 

AC = -h\ 2 (f>/ 2 (9-217) 

PC = -h' k 2 <t>/2n = ACIn (9-218) 

DC = 0 (9-219) 

LchC = —y 2 <f)lu' k 2 V (9-220) 

TchC = 0 (9-221) 

SSC = —y 2 (f>P/u' k 2 V (9-222) 

The symbols not previously defined are G t through G 8 , V, P, and SSC. The quantity 
V is the Abbe V number, the reciprocal relative dispersion of the material given by 

V= (n- 1)/An (9-223) 

The quantity P is the partial dispersion, given by 

P = (n,» — n r )/&n (9-224) 


where n r , n m , and n v are the indices at the long, middle, and short wavelengths, respec¬ 
tively, and A n = n v — n r as before. SSC is the contribution to the secondary spectrum 
of the system. Secondary spectrum is the residual longitudinal chromatic aberration 
when the foci for r and v light are equal; that is 

SSC = l' m ~ l' r = I'm ~ I'v (9-225) 


The terms G i through G 8 are functions of the index of the element. (The thin-lens 
third-order equations are often called "G-sums.”) 


G, = n 2 (n - l)/2 
G 2 = (2n + l)(n - l)/2 
G 3 = (3n 4- l)(n — l)/2 


G 5 = 2(n 4- 1)(n — l)/n 
G 6 = (3n + 2)(n — l)/2n 
G 7 = (2n + l)(n - l)/2n 


(9-226) 


G 4 = (n + 2)(n - l)/2n G 8 = n(n - l)/2 


404 


OPTICS 


To apply the thin-lens aberration expressions, the contribution equations (9-208 through 
9-214) are evaluated for each element of the system from the data of the thin-lens ray 
trace. The aberration at the image is then the sum of the contributions of all the 
elements. 

For analytical work the thin-lens contribution equations are often set up so that the 
aberration contribution is expressed as a function of a parameter whose value is to 
be determined. For example, in a two-element system, the spherical and coma con¬ 
tributions might be expressed as functions of the curvature of the first surfaces of 
the elements (ci and c 3 ), giving 


XSC = SCa 4- SC B — aci 2 4- bci 4- d 4- ec 3 2 -I- /c 3 -t- g 
SCC = CC A -I- CC B = hct + j + kc 3 4- m 


(9-227) 


if both elements are assumed to be in contact with the stop. If a zero value for the 
thin-lens third-order contribution were desired, the simultaneous solution of the two 
equations below would yield the necessary values of c i and c 3 : 


SSC = 0 = aci 2 4- be i 4- ec 3 2 -f /c 3 4- {d 4- g) 
SCC = 0 = hci -f Kca 4- (J 4- m) 


(9-228) 


9.6.3. Stop Shift Theory. The aberration contribution of an element depends upon 
its position relative to the stop or pupil of the system. 

9.6.3.1. Spherical Aberration. Spherical aberration is not affected by the position 
of the stop. It is a function of the effective size of the aperture, but the pupil may 
be placed anywhere in a system without changing the spherical aberration. 

9.6.3.2. Coma. Coma is affected by a change in stop position if there is spherical 

aberration in the system. This is indicated (although not proved) by Eq. (9-209). 
The value of Q is a function of y p (the height at which the principal ray strikes the 
element), which in turn is a function of the stop position. If the system has no spherical 
aberration, the stop position has no effect on coma. 

9.6.3.3. Field Curvature. The stop position has no effect on the Petzval curvature 
of a system but does affect the astigmatism and thus the field curvature, if coma or 
spherical aberration are present, as indicated by Eq. (9-210). The astigmatism of 
a thin element at the stop is a function of the element power only and cannot be changed 
by bending. 

9.6.3.4. Distortion. Distortion is affected by the stop position; a thin element in 
contact with the stop has no distortion contribution. 

9.6.3.5. Chromatic Aberration. Longitudinal (axial) chromatic is not a function 
of stop position, whereas lateral color is a function of the stop position. A thin ele¬ 
ment at the stop has no lateral color contribution. 

9.6.3.6. Example 1. Figure 9-25 shows a thin positive meniscus element. The 
portion of the element used by an oblique bundle of rays is moved further from the 
optical axis as the stop is moved farther from the lens. The rim ray curve of this lens 
is sketched in Fig. 9-26. The effect of shifting the stop along the axis is to select a 
different portion of the rim ray curve. With the stop in position A, the rim ray curve 
indicates outward flaring coma and inward curving tangential field. In position C 
the coma is inward flaring and the field is inward curving. Position B yields a system 
which is free of coma and which has a slightly backward-curving tangential field. 


THIRD-ORDER ABERRATIONS 


405 


This diagram illustrates how spherical aberration and coma are related, and also 
illustrates a basic law of stop shift theory — in the presence of undercorrected spherical 
aberration the position of the stop which eliminates coma (the "natural” stop position) 
also produces the most backward-curving field possible. 



Positions 

Fig. 9-25. Thin positive meniscus element. 



H' 


9.6.3.7. Example 2. Figure 9-27 shows a symmetrical erector system of the type 
used in terrestrial telescopes to invert the image formed by the objective. Each doublet 
is shaped to be free of spherical aberration; thus the size of the space between them, 
which determines the stop position, has no effect on the coma of the system. However, 
each doublet is designed to have a sizable amount of coma. Thus as the spacing be¬ 
tween the doublets is changed, the astigmatism of the system is varied. The spacing 
is usually chosen so that the astigmatism is either zero or slightly positive so as to 
"artificially” flatten the field curvature. 


Diaphragm 



Fig. 9-27. Symmetrical erector system to invert image formed by objective. 

































406 


OPTICS 


9.6.3.8. The Symmetrical Principle. Example 2 is successful because the coma 
in each element is canceled by the coma in the other. In an optical system which is 
completely symmetrical about the stop, there is no coma, distortion, nor lateral color. 

The provision of complete symmetry requires that the system work at unit mag¬ 
nification to be completely effective. However, even at infinite conjugates the coma, 
distortion, and lateral color are usually reduced to negligible values. 

9.6.4. Afocal Systems. The expressions given above for third-order aberrations 
become indeterminate for afocal systems, in which u\ is zero. Only a general outline 
of the necessary modifications to the equations is presented. (See Sec. 10.1.1 for a 
detailed discussion of afocal systems.) 

In an afocal system, aberrations are best described in their angular form (See Sec. 
9.4.1). Transverse aberrations (TA) and longitudinal aberrations (LA) can be con¬ 
verted to angular aberrations by use of the following equation: 

AA = TAu' k / yi = LA wV/yi (9-229) 

The aberration contribution expressions of the preceding paragraphs can be modified 
in this manner to an angular aberration form; this will eliminate their interdeterminacy 
by canceling the u\ which is found in one form or another in the denominators of each 
of these expressions. The following are longitudinal aberrations: SC, AC, PC, LchC, 
SSC. The following are transverse aberrations: TSC, TAC, TPC, TLC, CC,DC,TchC. 

9.7. Optical Design Techniques 

Optical design involves solving a number (say n) of second- (or higher) -order equa¬ 
tions in m variables, where n represents the number cf aberrations or characteristics 
which must be controlled and m represents the number of effective parameters that 
the designer has available for manipulation. 

9.7.1. General Considerations. Primary requirements to be imposed on the optical 
system, aperture, focal length, and field coverage, and specialized requirements such 
as length of working distance must be determined. The resolution or definition neces¬ 
sary, and the spectral bandwidth must also be considered. 

In infrared work, the choice of optical systems is usually between refracting and 
reflecting systems, depending on application, the materials acceptable to the applica¬ 
tion, and the necessity for chromatic correction. 

9.7.2. Correction of Primary Aberrations. After the type of optical system has 
been selected or invented, the next major step in the design process is the correction 
of the primary aberrations, or at least the correction of as many of them as are necessary 
and feasible. 

9.7.2.I. First Steps. The elements must be arranged to provide the desired optical 
characteristics, such as focal length, aperture, field, etc., for the system. 

The usual method for correction of aberrations is bending of the elements. The longi¬ 
tudinal chromatic aberration, lateral color, Petzval curvature, and to a certain extent 
distortion are unaffected by bending. Chromatic aberration and Petzval curvature 
must be corrected in the original power and space layout. The thin-lens contribution 
equations (see Sec. 9.6.2) are useful at this stage, and it is ordinarily a relatively 
straightforward procedure to adjust the system so that the XLchC, STchC*, and 2PC 
are equal to values which have been selected as desirable. 

Then the spherical aberration, coma, astigmatism, and distortion must be corrected 
to their desired values. It is probably best at this stage to make a graph of the aberra¬ 
tion contributions from each element as a function of the element shape. From a set 


OPTICAL DESIGN TECHNIQUES 


407 


of such graphs a region (or regions) for the solution can be selected. These graphs 
can be made from data obtained by the use of the thin-lens contribution equations (see 
Sec. 9.6.2), the surface contribution equations (see Sec. 9.6.1) or, in certain cases, by 
direct ray tracing. The last two procedures are more appropriate for work with elec¬ 
tronic computers. 

When the region of the solution is selected, a method of differential correction is 
applied. The partial differentials of the aberrations against shape are computed and 
also the value of the aberrations for a trial prescription. The desired amount of change 
of each aberration (A A) is determined by analysis of the trial prescription; the necessary 
simultaneous equations of the form 

i=k zA 

aa = 2 -ac, 

i = 1 

are set up and solved. Because of the nonlinearity of the equations, the solution is 
seldom precise; however, the preselection of the solution neighborhood limits the size 
of A C so that the simultaneous solution is a good approximation and a series of solutions 
converges rapidly on the desired design shape. 

9.7.2.2. Limiting the Parameters. If three aberrations A, B, and C are to be cor¬ 
rected by adjustment of three parameters x, y, and z, an initial trial of x, y, and z can 
be modified by changing one of the parameters, say z, so that one of aberrations, say 
C, is "corrected.” Then parameter y is changed and a new value of z is determined 
to hold the correction of C. Parameter y is varied in this manner until aberrations 
B and C are simultaneously corrected. Then parameter x is changed and, with each 
change of x,y, and 2 are adjusted as above to hold aberration B and C as desired. Then 
x is varied in this manner until aberration A is brought to correction simultaneously 
with B and C. Graphs of C vs. z, B vs. y, and A vs. x are useful in such a procedure. 

9.7.2.3. Adding Thickness. If the thin-lens expressions have been used in the 
preceding steps, it is necessary to add thickness to the elements. This is generally 
done by adjusting the secondary curvature of each thick element to hold the thick- 
element power equal to the thin-lens element power. The spacing between elements 
is then adjusted so that the separation of the principal points of the thick element 
is equal to the thin-lens spacings. This method serves to retain the overall system 
power and working distance at the same values as the thin-lens system. 

9.7.2.4. Trigonometric Corrections. When the aberrations have been corrected 
using third-order aberration contributions (either thin lens or surface contributions), 
it is necessary to trace rays trigonometrically to determine the actual state of correction 
of the system. It will usually differ by a small amount from that predicted by the 
third-order expressions. However, a step or two of differential correction as outlined 
in Sec. 9.7.2.1 will usually bring the trigonometrical correction to the correct value. 

It is possible to go directly to trigonometric correction from the thin-lens expressions 
(provided that the method of introducing thickness is rigorously consistent). Alterna¬ 
tively, an additional step of determining and correcting the surface contributions may 
be desirable after thicknesses are introduced. 

9.7.3. Reduction of Residual Aberrations. After the primary aberrations have 
been brought to correction, the design is usually tested for residual aberrations. The 
primary aberrations are generally corrected for a single zone of the aperture or field and 
can be expected to depart from correction in all other zones, as discussed in Sec. 9.4.5. 

If there were any parameters that were not used in the correction of the primary 
aberrations, these may be systematically varied and their effects on the residuals 



408 


OPTICS 


noted and used. The possibility that more than one "neighborhood of solution" exists 
should not be overlooked; it is, in effect, an extra parameter. 

An analysis of the source of the third-order contributions will often pinpoint one 
or two especially heavy contributory elements of the system. A reduction of a single 
large contribution will often reduce residual aberrations. This can be accomplished 
by introducing a correcting element near the offender (for example, convert a single 
positive element into a positive-negative doublet) or by splitting the offending element 
into two elements whose total power equals that of the original. This latter technique 
introduces two new variable parameters; the ratio of the powers of the two new elements, 
and the shape of the added element. Alternatively, a new shape for the offender may 
reduce its contribution to an acceptable level. 

Where residuals are a problem, it is wise to reconsider the starting power and spacing 
layout. It is sometimes possible to revise the layout in such a way that the powers 
of the elements can be reduced. This is a rapid way of reducing residuals. 

9.7.4. Automatic Design. The electronic computer has made possible the auto¬ 
matic implementation of certain of the preceding steps; for example correction of the 
third-order aberrations to a desired set of values. In this technique the computer 
calculates the partial differentials of the aberrations with respect to the available 
parameters and solves the resulting simultaneous equations to determine the required 
changes necessary. These approximate changes are then automatically applied to 
their respective parameters, and the process is repeated until the aberration contribu¬ 
tion sums are within predetermined limits of the desired values. The system is then 
submitted to a trigonometric ray-trace check, and the process is repeated if necessary 
until the ray-trace aberrations are corrected. 

Another school of automatic correction uses a "merit function,” which is typically 
a weighted average of the absolute departures of the intersections of many rays from 
an ideal point image. Various techniques (least squares, steepest descent, and others) 
are used to improve the "merit function” automatically. 

It is not possible in a handbook of this size to provide the automatic lens-design 
programs. From the references cited in the summary below, one can obtain some 
of the programs and get in touch with those responsible for their development or use. 

Two distinct types of automatic lens correction are evident in the published literature. 
The first code gives the problem to the computer in explicit mathematical terms, thus 
making it possible for any engineer or scientist, with a modest knowledge of optics, 
to obtain lens designs [1-4]. The second type requires direction by a skilled specialist 
who makes qualitative judgments and compromises; hence the computer should be 
regarded as a tool presenting the designer with provisional solutions only [5,6]. Work 
is also being done in the design and manufacturing of aspheric optical elements [7,8]. 
Bell and Howell have been quite active in this branch of design and have developed 
a computer-programmed lens-grinding system [9]. 

Procedures and typical designs are presented in a thesis by G. Spencer [10], from 
which we quote directly: 

In 1954, Rosen and Eldert described a method designed to reduce the values of a 
large number of ray deviations — a number in excess of the number of available 
variables [11]. This led them to a least squares formulation. Hopkins, McCarthy, 
and Walters [12] and McCarthy [13], on the other hand, were interested in arriving 
at specific values for the first-order chromatic and third-order monochromatic 
aberrations — altogether seven in number and less than the number of available 
variables. This led them to a modified Newton-Raphson procedure, described 
by them in 1955. Feder phrased the problem in terms of the reduction of a single 


ACHROMATISM AND ACHROMATS 


409 


"merit function” for the system with the implication that the smaller the value 
of the merit function the better the state of correction of the system. This led 
him to an extensive investigation of various gradient methods which he published 
in 1957 [14]. Modified gradient methods which maintain constant values for 
certain system characteristics while simultaneously reducing the value of a merit 
function have been described both by Feder [14] and by Meiron and Lobenstein [15]. 

The methods of Rosen and Eldert and of Hopkins and McCarthy may be termed 
linearization methods since they involve the approximation of non-linear functions 
by linear ones at each iteration. Practically all of the procedures thus far reported 
in the literature may be classified as either linearization methods or gradient 
methods. Procedures which defy these classifications have been described by 
Black [16] and by Meiron and Volinez [17]. These methods involve the successive 
adjustment of individual system parameters to values which minimize a figure 
of merit for the system. They may be called relaxation methods. Black also 
mentions the use of block and group operations on system parameters, which are 
standard relaxation techniques. Relaxation methods appear to be considerably 
less efficient than either gradient or linearization methods, however. 

Probably the most successful automatic correction methods to appear thus far 
are the SLAMS method introduced by Wynne [18] and Nunn and Wynne [19] 
and the conjugate gradient method investigated by Feder [20]. 

Other references include the work of Holladay [21], of Gray [22], and of O’Brien [23]. 

9.8. Achromatism and Achromats 

Many infrared systems make use of reflecting optics because of their freedom from 
chromatic aberration. 

The condition for achromatism can be taken directly from the thin-lens aberration 
contribution equation (9-220) and is given by 


SLchC = 0 = - 


U \ 2 


Xy 2 (f>/V 


(9-230) 


For a system of two refracting elements in close contact, the powers of the elements 
may be solved for directly, giving 


4>a — 


(V A - Vs) 


(<t>AB — RV b) 


(9-231) 


(f>B ~ 


R = 


(Vb - Va) 


(4>ab ~ RV a) 


LchA LchA' 


l 2 


l ' 2 


_ / ly _ ( l v l r \ 

= V l v lr / ” v I'vl'r I 


(9-232) 


(9-233) 


where <f>A, 4>b, <\>ab are the powers of the elements A and B and the doublet AB, re¬ 
spectively, V A and V H are the Abbe V numbers ( n m — 1 ln r — n r ) of the elements A and 
B, and R is a residual chromatic aberration term as defined in Eq. (9-233). In the 
Eq. (9-233), LchA is the chromatic aberration of the object a distance l from the doublet 
and LchA' is the chromatic desired in the image located a distance /' from the doublet: 
l v and l r are the object distances in short and long wavelengths, respectively, and 
l' v and l'r are the image distances. If a real object is to be imaged without any chro¬ 
matic aberration (as is usually the case) the equations reduce to 









410 


OPTICS 


<t>A = V A <J> A s/(V A - Vb) (9-234) 

(f) B = V B (I> A b/(Vb V a ) (9-235) 

In the infrared, it is necessary to check the secondary spectrum carefully. For example, 
ordinary optical glasses are often used in the lead sulfide region and one might calculate 
an "achromat” from Eqs. (9-234) and (9-235) using F-values based on wavelengths of 
1.0 fx, 1.8 ix, and 2.5 ix which would yield a doublet that brought radiation of 1.0 ix and 
2.5 ix to a common focus. However, on ray tracing the intermediate wavelengths 
one would find a very large secondary spectrum, because for every pair of optical glasses 
there is a wavelength between 1.0 ix and 1.5 ix at which their reciprocal relative dis¬ 
persion (V* values) is identical. At this wavelength the doublet is no better corrected 
than a single element. To achieve chromatic correction over this particular spectral 
region it is necessary to use fluorite (CaF 2 ) in combination with a dense barium crown 
or a light flint glass. 

The thin-lens expression for secondary spectrum (Eq. 9-222) indicates a technique 
for handling this problem. For the secondary spectrum to be zero, the conditions 
that SLchC = 0 and XSSC = 0 are necessary. In a thin doublet 


2.SSC 




P. i</> 


A P B<t>B ^ 

V B ) 


(9-236) 


and if values for <f> A and </> B from Eqs. (9-234) and (9-235) are substituted into this 
expression, the following equation can be derived: 


1SSC = f(P B ~ Pa)/(V a ~ V B ) 


(9-237) 


in which P A must equal P B to achieve a zero secondary spectrum. A plot of P against 
V for the available materials (P and V calculated for the spectral band of interest) is 
useful in selecting the pair of materials with the smallest value of ( P B —Pa)/(V a — V B ) 
and hence the smallest secondary spectrum. 

9.9. Resolution, Definition, and Image Spot Size 

The resolving power of a system is the smallest angular separation of two equally 
bright point sources at which the system can detect that there are two sources rather 
than one. The resolution does not fully describe the performance of an optical system, 
and other criteria such as energy distribution and frequency response have come into 
use. The energy distribution is often represented by a plot of the percentage of the 
total image energy falling within a circle of a given diameter against the diameter of the 
circle. Frequency response is the percentage modulation of intensity in the image of 
an object of a given spatial frequency and is usually presented as a plot of percentage 
modulation or response against spatial frequency. 

9.9.1. The Airy Disc. The wave nature of light limits the characteristics of the 
image formed, even by an optical system without aberration, to a disc of illumination 
surrounded by rings of illumination which are progressively fainter for larger rings. 
This pattern is shown in Fig. 9-28. Figure 9-29 indicates the illumination levels in 
the pattern. (The illumination levels in the rings are highly exaggerated for clarity.) 

For a circular aperture,* the illumination distribution in this pattern is given by 


E = Kp 4 


1 



(9-238) 


*See Fig. 9-30 for meaning of symbols. 









RESOLUTION, DEFINITION, AND IMAGE SPOT SIZE 


411 



Fig. 9-28. Appearance of diffraction pattern. 


Illumination 



Fig. 9-29. Relative illumination in Airy disc with the size of the 
central maxima reduced in scale. 



Fig. 9-30. Meaning of symbols in Eqs. (9-238) through 
(9-242) and Table 9-2. 










412 


OPTICS 


in which 

m = ^‘ 7Tf) s i n a (9-239) 

\ 

where E = the illumination at point Z 

p = the semidiameter of the aperture of the system 
X = the wavelength of the energy 

a = the angle subtended by the distance from the point Z to the axis, from the 
aperture 

For a slit aperture the pattern becomes a series of lines and the illumination is given 
by 

Cl n 2 »yj 

E = K - (1 + cos a) (9-240) 

m 2 

For a rectangular aperture,* the illumination is 

sin 2 m i sin 2 m 2 

E= K Pl 2 P 2 2 ---— (9-241) 

m i 2 m 2 2 

Table 9-2 gives the location, relative illumination, and integrated illumination for 
the central disc and the first four rings for the circular and slit apertures. 

The Rayleigh criterion assumes that two points are just resolved if the central max¬ 
imum of one image pattern is directly over the first dark ring of the other, and vice 
versa. Figure 9-31 illustrates this condition. Thus (from Table 9-3) the separation 
for this condition is 


Z = 0.61X//1 sin 6 (9-242) 

where Z is the linear separation between the images, X is the wavelength, n is the final 
index, and 6 is the half-angle subtended by the aperture from the image. 


Table 9-2. Location, Relative Illumination, and Integrated 
Illumination for Circular and Slit Apertures 

Circular Aperture Slit Aperture 


Ring 
(or Band ) 


Central max. 
ls£ dark 
2nd bright 
2nd dark 
3rd bright 
3rd dark 
4th bright 
4th dark 
5th bright 
5th dark 


Z 

(X/n sin 6) 

0 

0.61 

0.819 

1.116 

1.333 

1.619 

1.847 

2.120 

2.361 

2.622 


Peak 

Illumination 

1.0 

0 

0.01745 

0 

0.00415 

0 

0.00165 

0 

0.00078 

0 


Total Ring 
Energy 
Relative to 
Central Disc 

1.0 

0.084 

0.033 

0.018 

0.011 


Proportion 
of T otal 
Energy in 
Each Ring 

0.839 

0.071 

0.028 

0.015 

0.01 


Z 

(X/n sin d) 

0 

0.5 

0.715 

1.0 

1.230 

1.5 
1.736 
2 

2.235 

2.5 


Peak 

Illumination 

1.0 

0 

0.0469 

0 

0.0168 

0 

0.0083 

0 

0.0050 

0 


*See Fig. 9-30 for meaning of symbols. 









RESOLUTION, DEFINITION, AND IMAGE SPOT SIZE 


413 



Fig. 9-31. Rayleigh criterion for resolution; 
peak intensity from one Airy disc dislocated 
at first dark ring of other Airy disc. 


If the optical system is reasonably aplanatic, p/l' can be substituted for n sin 6 (if 
the image is in air) and the following expression derived for the limiting angular 
resolution ( Z/l ') of an optical system 


0=0.61 kip (9-243) 

where 0 = the angular separation of the two objects which are just resolved 

p = the semiaperture of the optical system 

9.9.2. The Effects of Aberrations on the Airy Disc. Rayleigh established a 
criterion for the amount of aberration which could be tolerated without "sensibly” 
degrading the image of an otherwise perfect system. The Rayleigh limit can be ex¬ 
pressed as follows. 

An image can be "sensibly” perfect if there exists not more than one quarter-wave- 
length difference in optical path over the wavefront with reference to a spherical 
wavefront about the selected image point (see Tables 9-3 and 9-4). 


Table 9-3. Distribution of Energy in Central Disc 


Amount of Aberration 

Perfect system 
1/4 Rayleigh limit 
1/2 Rayleigh limit 
1 Rayleigh limit 


Energy in Central Disc 

(%) 

84 

83 

80 

68 


Energy in Rings 
(%) 

16 

17 

20 

32 


Table 9-4. Rayleigh Limit in Geometric Terms 


One Rayleigh limit of: 

Out of focus = 

Third-order longitudinal spherical aberration = 
Residual zonal longitudinal spherical aberration = 
Sagittal coma = 

Longitudinal chromatic aberration ~ 


k/2n' sin 2 U' m 
4k/n' sin 2 U' m 
6k/n' sin 2 U' m 
k/2n' sin U' m 
k/n' sin 2 U' m 


9.9.3. Geometrical Limits on Resolution. When aberrations are small, their 
effects on image size should be evaluated in terms of the diffraction pattern. However, 
when the aberrations are quite large, the blur spot predicted by ray tracing may be 
used to determine the image characteristics. 




414 


OPTICS 


9.9.3.I. Third-Order Spherical Aberration. Figure 9-32 shows the pattern of ray 
intersections at the image of a lens system with pure third-order spherical aberration, 
that is, aberration which can be described by the equation LA — ay 2 . There is a 
well-defined minimum diameter blur spot, which occurs at 0.75 LA m from the paraxial 
focus. At this point the blur diameter is given by 

B m i n = 0.5 LA m tan U' m (9-244) 

and is one-fourth the size of the blur at the paraxial focus. 

When the aberration is small, the position of best focus (chosen on the basis of wave- 
front aberration or minimum optical path difference, O.P.D.) is halfway between 
the mariginal focus and the paraxial focus. 


Focus for 



Fig. 9-32. Pattern of ray intersections at image lens system with 
pure third-order spherical aberration. 


9.9.3.2. Third- and Fifth-Order Spherical Aberration. In the presence of third- 
and fifth-order spherical aberration ( i.e ., when LA = ay 2 + by 4 ), there are two possi¬ 
bilities for optimization. On the basis of O.P.D. the best image occurs when the lens 
is designed so that LA m = 0, but the minimum geometrical blur circle occurs when 
LA Z =1.5 LA m . Both cases are illustrated in Fig. 9-33. 

Marginal spherical equal to zero. The minimum O.P.D. focus is 0.75 LA Z from the 
paraxial focus but the minimum diameter blur occurs at 0.422 LA Z from the paraxial 
focus. At this latter point the blur diameter is given by: 


Bmin 0.84 LA Z tan U' m (for small angles) 

B m in ~ 0.42LA* (tan U'm + sin U' m ) for larger angles 


(9-245) 













RESOLUTION, DEFINITION, AND IMAGE SPOT SIZE 


415 


-Focus Positions for Minimum O.P.D. 

and Minimum Blur Spot Diameter 


Axial Intercept 


of Rays 



Ray Height 


m 


0.707 y 


m 


Fig. 9-33. Geometrical image formation in presence of third- and fifth-order spher¬ 
ical aberration, (a) where LA m = 0 and (6) where LA Z = 1.5 LA m (state of correction 
actually produces smaller geometrical blur spot although O.P.D. is less in case where 

LA m = 0). 


Marginal spherical equal to 0.67 zonal spherical. The minimum O.P.D. focus is 
0.75 LAmax from the paraxial focus but the minimum diameter of the blur spot is 
1.25 LA m from the paraxial focus and is given by 


B min ~ 0.5 LA m tan U' m (9-246) 

9.9.3.3. Chromatic Aberration. The treatment of blur-spot size in the presence 
of longitudinal chromatic aberration is complicated because the spectral response 
of the detector is a factor. Figure 9-34 shows the image energy distribution under 
these circumstances. The insert figures show the spectral response plots. The main 
graphs plot the percentage of the total energy within a given diameter area, where 
the diameter of the area is given as a fraction of the total size of the chromatic blur 
spot. The total size of the chromatic blur spot can be determined from 


Bmin = LchA • tan U' m = (/'» — l \) tan U' m (9-247) 

The energy density in the chromatic blur spot is not uniform; 75 to 95% of the "effec¬ 
tive” energy is contained within the central half diameter (or quarter of the area) 
of the spot. 

It is often of interest to determine the image characteristics at the image of a sharp 
edge. The intensity at the image of an edge in the presence of chromatic for the two 
response characteristics is shown in Fig. 9-35. 



















Fraction of Total 

Energy Within Diameter 


416 


OPTICS 



Diameter of Area ■ 
as a Fraction of Total 
Chromatic Blur Size 

Fig. 9-34. Energy distribution in presence of longitudinal chromatic 
aberration with plane of reference midway between extreme focii. 
Graph indicates percentage of total energy in image that falls within 
a circle of given distance. 


e 

o 


c 


S 

3 


0> 

> 


a! 


K 


1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

0 


r 


Geometrical Position 
of Image of Knife Edge 



m 


Distance from ’Ideal" Position of Knife Edge Image 
(as a fraction of the total chromatic blur size) 


Fig. 9-35. Intensity gradient at image of knife edge 
in presence of longitudinal chromatic with plane of 
focus midway between extreme focii. 






SUMMARY OF EQUATIONS 


417 


9.10. Summary of Equations 


n' sin 1’ — n sin I 

_1 = _1 1 
s' s + f 

f 2 = -xx' 

f -x' 

m = = — 

x f 


I = hnu = h' n' u' = ynu pr 

. 1 , ri 1 (n- 1)M 

power = (/>= — = (n — 1)-1- 

f Lr, r 2 nr,r 2 J 


= (n - 1) [ci — 


, (n-1) , 

c 2 H- tCiC 2 

n 


u'i — Ui + yi(f>i 
yt+i = yt — d'iUi 
fAfs 


fAB — 

fA = 

f B = 


fA+fB—d 

fABd 


fAB ~ S’ 

—s' d 


fAB — s' — d 


, /sS' ^ ^ f A fB (V + f A ) ± V (v — f A ) 2 - 4f B (v — f A ) 

d — — ; — J a + Jb — 


fB — s' 
X</> = min 


f 


AB 


d — V — f A B — V fAB(f AB v) 

= min 


s' — /^ 4 B ± "\/ fA B A B 
(u'irt'i ) = Wi-in,- + y ,(n'j — n t )/r,- 
y«+i = yi ~ t'i(u'in'i)/n'i 
numerical aperture = N.A. = n sin u 

T A TA AA ' f 
LA tan U’m tan U' m 

LA' =L' -V 


coma/ = 

O.S.C. 


H'ab ~ H' pr 

comas _ sin U t 
h u i 


u k 


V - V 


Pr 


sin U'k ( L' — I'pr ) 


- 1 




















418 


OPTICS 


X s = L'cg ~ l' 

Xt — L ' ae — l' 

Distortion = H' pr — h' 
LchA = T v — l' r 
TchA = H'x) — H' r 
C.D.M. = ( H' v -H’ r )/h' 
LA' = L' - l' 

TA' = LA' tan U' 


_ sin U x U' ( l'-l'pr ) 

u ! sin U'’(L'-l' P r) 

x ptz = 1.5x s — 0.5x f 
Dist = H'pr — h' 

LchA = l'p l'r 
TchA = H' v - H'r 
Desk Calculations 

Paraxial Ray Trace: Sec. 9.5.3.1 
Meridional finite radii: Eq. (9-73) 

Meridional planes: Eq. (9-83) 

Computer Calculations 

Meridional: Eq. (9-101) 

Skew ray, spherical surface: Eq. (9-128) 

Skew ray, aspheric: Eq. (9-144) 

Third-order surface aberrations: Eq. (9-160) 
Third-order thin lens aberrations: Eq. (9-203) 

2LchC = 0 = - Xy*<t>IV 
U k Z 

Va 

(b 4 =--- ((b AB - RVb ) 

(V A ~ V B ) 


~ 77} 7T7 (0.4B ~ R Va ) 

\V B v A) 


LchA 

LchA' / 

f lv ~ lr\ 

(l'v -l'r\ 

/ 2 

l>2 \ 

V l vlr ) 

V I'vl'r ) 


4>A — Va<{)AbI(V A ~ V b) 

XSSC = f(P B -PA)/(V A -V B ) 


E = Kpi 2 p 2 2 


sin 2 m i 

mi 2 


sin 2 m 2 

m 2 2 


Z = 0.61 \/n sin 6 
(3 = 0.61 X/p 














References 


BIBLIOGRAPHY 


419 


1. C. A. Lehman, LASL Design Program for the IBM 7090, Los Alamos Scientific Laboratory 
Report LA-2837 (1963). 

2. B. Brixner, Lens Designing Technique with the 1962 LASL Code for the IBM 7090, Los Alamos 
Scientific Laboratory Report LA-2877 (1963). 

3. D. P. Feder, J. Opt. Soc. Am. 52, 177 (1962). 

4. Unpublished program of Illinois Institute of Technology Research Institute, Chicago, Ill. 

5. R. E. Hopkins, Research on Fundamentals of Geometrical Optics (ORDEALS FOR IBM 7070), 
The Institute of Optics, University of Rochester, Contract AF-49(638)-668, Project 976, Task 
37650 (1962). 

6. R. E. Hopkins and G. Spencer, J. Opt. Soc. Am. 52, 172 (1962). 

7. G. W. Shavlik, Modifications and Additions to the NOTS General Optical Ray Tracing Com¬ 
puter Program, U.S. Naval Ordnance Test Station, China Lake, Calif., Report 7966 (1962). 

8. J. R. Snyder, Photophysics Branch, Instrument Development Division, U.S. Naval Ordnance 
Test Station, China Lake, Calif. 

9. D. Stevens and J. Thiel, AF 33(600)-37199 Bell & Howell Company, Chicago, Ill. 

10. G. Spencer, Thesis, College of Engineering and Applied Science, University of Rochester, 

Rochester, N.Y. 

11. S. Rosen and C. Eldert, J. Opt. Soc. Am. 44, 250 (1954). 

12. R. E. Hopkins, C. A. McCarthy, and R. Walters, J. Opt. Soc. Am. 45, 363 (1955). 

13. C. A. McCarthy, J. Opt. Soc. Am. 45, 1087 (1955). 

14. D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957). 

15. J. Meiron and H. M. Lobenstein, J. Opt. Soc. Am. 47, 1104 (1957). 

16. G. Black, Proc. Phys. Soc. (London) B68, 729 (1955). 

17. J. Meiron and G. Volinez, J. Opt. Soc. Am. 50, 207 (1960). 

18. C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959). 

19. M. Nunn and C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959). 

20. D. P. Feder, J. Opt. Soc. Am. 52, 177 (1962). 

21. J. C. Holladay, Computer Design of Optical-Lens Systems (IBM 704 ) in Computer Applications — 
1960, Benjamin Mitteman and Andrew Ungar, (eds.) Macmillan Company,New York(1961). 

22. D. S. Gray, J. Opt. Soc. Am. 53, 672 (1963). 

23. Private communication from C. O’Brien of Illinois Institute of Technology Research Institute. 

Bibliography 

The literature does not abound with books written by lens designers. They seem to 
design lenses. Some useful books are: 

Born, M., and Wolf, E., Principles of Optics, Pergamon, New York (1959). 

Buchdahl, H. A., Optical Aberration Coefficients, Oxford University Press, London (1954). 

Conrady, A. E., Applied Optics and Optical Design, Vols. I and II, Dover, New York (1957 and 
1959). This is the book on which most lens designers first cut their teeth, even though they 
may now use somewhat different techniques and symbols. 

Handbook of Optical Design (MIL HDBK 141). This handbook, which contains much practical 
design information and very useful discussion of procedures, should soon be published by the 
Government Printing Office (written 9/10/63). The material was prepared primarily by staff 
members of the University of Rochester, Institute of Optics. It has also appeared in part as 
Summer Course Notes. Some copies are available from the Institute. 

Herzberger, M., Modern Geometrical Optics, Interscience, New York (1958). 



























































































































Chapter 10 

OPTICAL SYSTEMS 


Warren J. Smith 

Infrared Industries, Inc. 


CONTENTS 


10.1. Useful Optical Devices. 422 

10.1.1. Afocal Systems. 422 

10.1.2. Relay Systems, Erectors, and Periscopes. 424 

10.1.3. Aplanatic Surfaces and Elements. 427 

10.2. Detector Optics. 427 

10.2.1. Application. 427 

10.2.2. Field Lenses. 429 

10.2.3. Light Pipes (Cone Channel Condensers). 430 

10.2.4. Immersion Lenses. 433 

10.3. Refracting Objectives. 434 

10.3.1. The Single-Element Objective. 434 

10.3.2. Achromats. 436 

10.3.3. Combinations of Several Thin Single Elements. 436 

10.4. Reflecting Objectives. 437 

10.4.1. The Spherical Reflector. 437 

10.4.2. The Parabolic Reflector. 440 

10.4.3. Other Conics: The Ellipsoid and Hyperboloid. 441 

10.4.4. Compound, or Double, Reflecting Systems. 441 

10.5. Catadioptric Objectives. 443 

10.5.1. The Schmidt System. 443 

10.5.2. The Mangin Mirror. 444 

10.5.3. The Bouwers (Maksutov) Objective. 445 

10.5.4. Aberrations of a Plane Parallel Plate in a 

Convergent Beam. 447 

10.6. Rapid Estimation of Image Blur Size for Several Optical Systems .... 448 

10.6.1. Diffraction-Limited Systems. 448 

10.6.2. Spherical Reflector. 450 

10.6.3. Mangin Mirror. 452 

10.6.4. Parabolic Reflector. 452 

10.6.5. Schmidt System. 453 

10.6.6. Catadioptric Systems. 453 

10.6.7. Refracting Systems. 454 

10.6.8. Summary of Equations. 455 


421 



































10. Optical Systems 


10.1. Useful Optical Devices 

10.1.1. Afocal Systems. An afocal system has its object and image at infinity and 
thus has no focal length. It is composed of two or more components so arranged that 
(in a two-component system) the image of the first component, which is the object for 
the second, lies exactly at the first focal point of the second component and is thus 
reimaged at infinity. Figures 10-1 and 10-2 show afocal systems. 





Fig. 10-1. Astronomical, Galilean, and terrestrial telescopes. 


422 



































USEFUL OPTICAL DEVICES 


423 





Fig. 10-2. Afocal systems (lower sketches show reflecting equivalents 
or refracting systems above). 


10.1.1.1. Magnification , or Magnifying Power. An afocal system usually has a 
characteristic magnification, in that the object appears larger (or smaller) when viewed 
through the telescope than with the unaided eye. Figure 10-3 shows a simple tele¬ 
scope with an objective focal length of f\ and an eye-lens focal length of f 2 . The object 
subtends a half-angle of cti at the objective; thus the image formed by the objective 
has a height equal to /i tan a,. The half-angle subtended by the internal image at the 
eye lens is thus the image height divided by the focal length of the eye lens, giving 


tan a 2 = — tan «i 
fi 


( 10 - 1 ) 


Since the apparent angular size of the object when viewed through the telescope is 
given by tan a 2 , the object will appear to have been magnified by f lf 2 , where the mag¬ 
nification is given by 

M=fjf 2 (10-2) 


The diameter of the axial bundle of rays emerging from the telescope is also governed 
by the magnification. From Fig. 10-3, 


CAi A 
CA 2 fi 


(10-3) 











































424 


OPTICAL SYSTEMS 


The space between objective and eye lens is equal to f\ + ft for both the astronomical 
and Galilean telescopes. Also, the eye lenses of the astronomical and terrestrial 
telescopes form an external image of the objective aperture, through which all the 
energy emerging from the telescope passes. This image is called the exit pupil and is 
the customary location of the eye (see Sec. 9.3 on stops and apertures). The Galilean 
telescope has a virtual exit pupil inside the telescope [1]. 

10.1.1.2. Anamorphic Systems. An anamorphic system has a focal length or mag¬ 
nification in one meridian different from that in the other. In Fig. 10-4 the eye is 
replaced by a lens of focal length f 3 at the exit pupil. The focal length of the combina¬ 
tion is equal to the magnification of the telescope times f 3 . If the telescope is composed 
of lenses with cylindrical surfaces, the overall system will have a focal length of M •f 3 
in one meridian, but in the meridian parallel to the cylinder axes it will have a focal 
length of f 3 because in this meridian the cylinders are acting as plane surfaces. A 
schematic anamorphic system is sketched in Fig. 10-5. 

Anamorphic systems can be used in infrared work where it is desired to use a square 
or circular detector and at the same time obtain a wide field of view in one meridian 
and a narrow field of view in the other. 



Fig. 10-4. Anamorphic system with eye replaced by lens of focal length 
f 3 at the exit pupil. 


10.1.1.3. Applications of Afocal Systems. Figure 10-2 shows the refracting and 
reflecting equivalents of astronomical and Galilean telescopes. These systems can be 
used in infrared work to reduce the diameter of the collected bundle of rays from the 
target. Figure 10-6 shows such an application. Because the field angles are increased 
by magnification at the same time that the diameter of the bundle of rays is decreased, 
this device is limited to systems with small fields of view. 

10.1.2. Relay Systems, Erectors, and Periscopes. The erector lens of the Ter¬ 
restrial telescope (Fig. 10-1) indicates the technique of inverting an image with a 
relay system, which may at the same time magnify or minify the image. To carry 
an image through a long narrow path, one may use a series of relay systems, with each 
subsequent system reimaging the original object. Figure 10-7(a) illustrates a triple 
relay system in which each relay lens operates at unit magnification. A field lens can 
be introduced at (or near) each image plane to redirect the energy from one relay 
lens into the aperture of the succeeding relay, as shown in Fig. 10-7(6). The field 
lens forms an image of the aperture of one relay system in the aperture of the next. 

10.1.2.1. Projection Systems and Condensers. A schematic projection system is 
shown in Fig. 10-8. The reflector is a spherical mirror positioned with its center 
of curvature at the light source so that it produces an inverted image of the source 
in the same plane and at the same size as the source. This raises the average effective 
brightness of the source, either by filling in the gaps or openings in the filament or 


















USEFUL OPTICAL DEVICES 


425 



Fig. 10-5. Schematic anamorphic system. 



Fig. 10-6. Application of a reflecting afocal system to two-color radiometer. 
Energy alternately traverses paths A and B. When the mirrored chopper 
blade is in the beam at X, it is not in the beam at Y, and vice versa. 







































































426 


OPTICAL SYSTEMS 



Fig. 10-7. Relay and field lenses: (a) A three-element relay system in which 
most of the light from lens A escapes past lens B and lens C. In (b) a field 
lens D is introduced at the image plane to direct light from lens A to lens B. 
The field lens is constructed to image the aperture of A into the aperture of B. 



Fig. 10-8. Schematic projection system. 


arc, or by raising the temperature slightly, or both. The condenser images the source 
in the aperture of the projection lens. For optimum efficiency, the image of the source 
should completely fill the aperture of the projection lens. The projection lens images 
the object at the desired position. 

If the light source is replaced by a detector, the projection system becomes a typical 
infrared device, with the projection lens, object, and condenser functioning as objective, 
reticle, and field lens, respectively. 

Condensers (and field lenses) are usually made up of one or more single elements. 
The power and position of the condensing system as a whole are calculated from the 













DETECTOR OPTICS 


427 


equations given in Sec. 9.2 to produce an image of the desired size and location. An 
estimate of the number of elements necessary is made, and the total power is divided 
into that number of parts. Each element is shaped for minimum spherical aberration, 
either by use of the third-order thin-lens contribution (Eq. 9-215), or by shaping them 
so that a marginal ray is deviated equally at each surface. The system is then tested, 
either by trigonometric ray tracing or by constructing a sample. If the residual spher¬ 
ical aberration is too large, the number of elements may be increased, or one or more 
surfaces may be aspherized. 

10.1.3. Aplanatic Surfaces and Elements [2,3] 

10.1.3.1. Aplanatic Surfaces. The spherical aberration contribution of a surface 
is zero in the following three cases: 

(а) When the object and image both lie at the surface (/ = /' = 0) 

(б) When the object and image both lie at the center of curvature of the surface 
(/=/' = r) 

(c) When the object lies at 

l = r{n' -I- n)/n (10-4) 

The following expressions also apply for this case: 

/' = r(n' + n)tn' (10-5) 

n' In — sin u' I sin u (10-6) 

Case c is the aplanatic condition. A cone of rays may be further converged by the 
use of an aplanatic surface without introducing any additional aberration. Cases 
b and c are both used in detector immersion lenses. 

10.1.3.2. Aplanatic Lenses [4-7]. A combination of cases c and b can be used to 
make a lens which will increase the convergence of a cone of rays without introducing 
spherical aberration. The first surface is an "aplanatic” surface of the case c type 
which converges the rays by the factor n'In of Eq. (10-6). The second surface is then 
made concentric with the image formed by the first surface. The rays are undeviated 
by the second surface, and the convergence of the entire cone of rays is increased by 
n'In, or, if the lens is in air, by its index. Figure 10-9 indicates the layout of such a 
lens. An aplanatic lens or surface can only increase (or decrease) the convergence 
(or divergence) of a cone of light. It does not bring parallel rays to a focus nor does 
it change a diverging cone of rays to a converging one, or vice versa. 

10.2. Detector Optics 

The optics immediately associated with infrared detectors are field lenses, immersion 
lenses, and light pipes (or cone condensers). All allow use of a smaller size detector. 
Field lenses and light pipes also redistribute the energy over the surface of the detector. 

10.2.1. Application. In Fig. 10-10 the objective is shown as a simple lens; however, 
the considerations outlined below apply also to a system with a compound objective 
or a reflecting-type objective. The objective has a focal length f and an effective clear 
aperture A. The detector size is D and the detector covers a half-angle of view a. 

If the objective is well corrected, then the relative aperture is given by 

/7no = fl A (10-7) 

and for moderate angles 

D = 2af (10-8) 

Substituting the value of f given by Eq. (10-8) into Eq. (10-7) gives 

//no = D/2aA 


(10-9) 


428 


OPTICAL SYSTEMS 



\ 

^ 


( 6 ) 

Fig. 10-9. Aplanatics: (a) aplanatic surface; (6) aplanatic lens 
where r x is an aplanatic surface, r 2 has its center of curvature at 
lx. This lens increases the convergence of the cone of rays by a 
factor of n without introducing aberration. 




Most infrared systems require a large aperture A to collect a maximum of radiation, 
a small detector D to optimize the detector signal to noise ratio, and a large field a. 
These requirements reduce the necessary ft no (that is, increase the "relative aperture”). 
The limit on the relative aperture of a well-corrected optical system is that it cannot 
exceed twice the focal length; that is, /70.5 is the smallest /Vno attainable (see Fig. 
10-11). In a well-corrected system, the Abbe sine condition must hold (p. 373). The 
sine condition can be expressed as 


Y = f sin u' 


(10-10) 






















DETECTOR OPTICS 


429 


Principal "Plane" 



Fig. 10-11. Principal "plane” of well-corrected system 
is a spherical surface. 


The limiting relative aperture is given by 

/■/no = f/2Y = f/A = 0.5 (10-11) 

If the /'/no from Eq. (10-11) is substituted into Eq. (10-9), the following theoretical 
limitation on the relationship of A, D, and a are obtained: 


D min aA (theoretical limit) 


( 10 - 12 ) 


In practice it is often undesirable to exceed a speed of ft 1.0, in which case the relation¬ 
ship becomes 


D min = 2 aA ("practical” limit) 


(10-13) 


This limit applies to any optical systems with the detector in air including systems 
with field lenses and systems with light pipes. Immersion of the detector in a material 
of index n allows a reduction in D by a factor of 1/ n. 

Equations (10-12 and 10-13) are among the first to be considered in designing an infra¬ 
red system. 


10.2.2. Field Lenses. Field lenses are used when a reticle is placed at the focus 
of the system, illuminating the detector evenly, or when a minimum-size detector is 
required for use with a primary objective of low relative aperture. Figure 10-12 
illustrates the principle of the field lens. The objective forms an image in the focal 
plane, beyond which the light rays diverge. The field lens causes the rays of light at 
the edge of the field to be bent toward the axis so that they all fall on the detector. 
When a field lens is designed, the power and position of the lens are selected so that 
the objective aperture imaged on the detector is the same size as the detector. Equa¬ 
tions (9-2) and (9-4) are used to determine the relationships between the quantities 
of Fig. 10-12: 


AID = SIS' 


1 


1 


+ 


1 


f (-S) S' 


(10-14) 


(10-15) 


Combining and solving for the focal length of the field lens gives 



SD 

(D + A) 


(10-16) 


The necessary diameter of the field lens can be deduced from the path of the extreme 
off-axis rays (shown dashed in Fig. 10-12). Vignetting these rays at the edge of the 















430 


OPTICAL SYSTEMS 



Fig. 10-12. Function of field lens in infrared system. 


field reduces the diameter of the field lens. The optimum location for the field lens 
is at the focal plane, where its diameter is minimized; practical considerations of 
interference with the reticle, or of the effects of small imperfections in the field lens 
upon system performance, usually enforce a compromise location. The illumination 
at the detector under these conditions is now identical to that at the aperture of the 
objective. If there is no vignetting, a point image may be moved all over the field of 
view without affecting the intensity or distribution of the illumination at the detector. 

Usually the objective aperture is circular and the detector is square. If the detec¬ 
tor’s signal-to-noise ratio varies inversely with the square root of its area, or directly 
as (1/Di), and if the optical limit of Eq. (10-11 and 10-12) has been attained with the 
image of the objective aperture forming a circle of illumination inscribed within the 
square detector, the detector can be reduced in size. A loss of only 10% in S/N occurs 
if the detector is reduced in size until its square is inscribed in the illumination circle. 

The field lens is designed to minimize aberrations and place the maximum amount 
of energy on the detector. The technique is to consider the field lens as an image- 
forming system in itself, and to evaluate (by ray tracing or third-order contributions) 
the quality of the image of the objective aperture which it forms. When high relative 
apertures are required of the field lens, the requirements for correction of coma, and 
to a lesser extent spherical aberration, usually lead to a two- (or more) element system, 
as shown in Fig. 10-13. The first element is often planoconvex and the second a menis¬ 
cus in a shape to minimize aberration and optimize the energy delivered to the detector. 



Fig. 10-13. Section through typical field lens 
used with infrared detectors. 


10.2.3. Light Pipes (Cone Channel Condensers) [8,91. A light pipe is a hollow 
(or solid) truncated cone (or pyramid), with highly reflecting walls, which collects 
light at its receiving end and channels it by successive reflections to the other end, 





















DETECTOR OPTICS 


431 


at which a detector is usually located. It serves the same function as a field lens 
and is subject to the same optical limitations on the relationship among system aper¬ 
ture, detector size, and system field of view (Eq. 10-7 through 10-12). A completely 
reflecting (hollow) light pipe can be constructed for situations where there is no suitable 
refracting material. Such a light pipe "scrambles” the imagery between detector 
and objective aperture. 

Figure 10-14 shows a section of a typical light pipe and the path of a ray through the 
pipe. Ordinarily the detector is located at the smaller end of the pipe and the image 
plane of the system is located near the large end. 



Fig. 10-14. Section through light pipe, with path 
of ray showing pipe function. 


Figure 10-15 shows the construction used for tracing the path of a ray through a 
light pipe. S i and S 2 are the upper and lower sides of the pipe, respectively. Since 
the image from a plane reflecting surface is located an equal distance behind the sur¬ 
face, the image of S 2 in Si is a line through the intersection of Si and S 2 , making the 
same angle (a) to S i that S 2 does. Subsequent reflected images of the sides are similarly 
constructed. Then a meridional ray through the system can be represented by a 
straight line, and the intersections of the line with the images of the surfaces are 
the locations of the reflections of the ray. In Fig. 10-15 the ray AB, which is shown 
by the straight line AB ', emerges from the pipe after three reflections. The ray rep¬ 
resented by AC, however, never reaches the small end of the pipe. It is turned around 
and eventually emerges from the large end of the pipe. The limiting case is a ray 
tangent to circle FG, which is the locus of the reflected images of the exit window of 
the pipe. 



Fig. 10-15. Graphic ray tracing through light pipe. 



432 


OPTICAL SYSTEMS 


If the large end of the pipe represents the field stop of an objective system and the 
small end the detector size, the minimum detector size is given by 

D min =aA (10-17) 

If x is the length of the light pipe, c the radius of the detector, and s the radius of the 
large end of the pipe, 


_ c s cos a 

s (c/s) — sin a 

There are a number of limitations on solid light pipes. If the sides of the pipe are 
made reflecting and the detector is in optical contact with the pipe (and of the same or 
higher index), the situation is much the same as that described above except that the 
rays are refracted toward the pipe axis upon passing through the entrance face. This 
increases the acceptance angle of the pipe by a factor equal to the index n, and the 
limiting size of the detector becomes 

D min =aA/n (10-18) 

Some solid pipes are made without any reflecting coating, and operate on the basis 
of total internal reflection. Total internal reflection (T.I.R.) occurs when light passes 
from a medium of higher index to a medium of lower index at an angle of incidence 
such that the angle of refraction is 90°. 

Figure 10-15 shows that the angle of incidence decreases as the ray passes down the 
pipe, thus establishing a limiting length on the pipe under these conditions. A con¬ 
struction similar to that of Fig. 10-15 can be made, except that at each surface the 
critical angle must not be exceeded. This limits the maximum acceptance angle to 

arc sin [(n ,/n ) 2 — 1] 1/2 

Another limit for solid pipes occurs if the detector is not in contact with the end of 
the pipe, or if the detector or its coverplate have a lower index than the pipe. Under 
these circumstances T.I.R. can occur at the end face of the pipe and prevent the light 
from leaving the pipe. 

The transmission efficiency of a light pipe is a function of the reflectivity of the 
surfaces, and if the pipe is solid, of the transmission of the material used. For merid¬ 
ional rays (as in Fig. 10-15) it is simple to determine the path length through the 
medium and the number of reflections for any given ray. The path length d is given by 

/ n x-l/2 

d = L(l — — sin 2 e) (10-19) 

where 6 = angle of incidence on front surface. The transmission is given by 

r = p "'/ 2 (1 - p 2 ) e~ Qd (10-20) 

where p = measured reflectance of the sides of the pipe; m, the number of reflections, 
is given by 

nL sin 6 

m =- 

2 r(n' 2 — n 2 sin 2 0) 1/2 

and r = radius of pipe. 

Light pipes are made in the form of truncated cones, cylinders, and pyramids (com¬ 
monly four sided). Figure 10-15 shows that from point A one sees a mosaic of reflected 
images of the detector laid on a spherical surface about point E. The light is "scram¬ 
bled” or "folded” over the detector surface and, if there is a reflection loss at the surfaces, 




DETECTOR OPTICS 


433 

provides a ready method of visualizing the decreased effective sensitivity at the edges 
of the detector mosaic. When the light pipe is conical, the analysis is more complex, 
because the first-order reflected images of the detector will blend into a single annular 
image, and each of the higher-order images forms its own annulus. 

When a light pipe is designed, convenient length and a detector size within the limits 
of Eq. (10-12) and (10-18) are chosen. A drawing of the system, in the manner of 
Fig. 10-15, is made, and a set of "typical” rays (usually the full aperture rays to an 
on-axis point and to a point at the extreme edge of the field) are drawn in. These rays 
are examined and adjusted for the limitations described above. The longer the light 
pipe, the greater the number of reflections and the greater the transmission path if 
the pipe is solid. 

10.2.4. Immersion Lenses. The immersion lens is used to decrease the detector 
size to obtain increased detector signal-to-noise ratios. The aplanatic surface described 
in Sec. 10.1.3.1. is used for this purpose. 

10.2.4.1. The Hemispherical Immersion Lens. The immersion lens with greatest 
applicability is the hemispherical element with the detector located at the center 
of curvature (see Fig. 10-16). In an optical system with the detector located either at 
the focus of the primary system (objective) or at the image of the objective aperture 
formed by a field lens, the energy focused on the detector converges with a half-angle 
u and the detector size corresponds to h, the maximum height of the image. Since 
this image is formed in air of index 1.0, the Lagrangian invariant is hu (Eq. 9-6). 
If a hemispherical lens is inserted in the system with its center of curvature at the 
focus of the rays, the convergence angle is not changed (since the rays of the axial 
cone strike the surface normally), and the Lagrangian invariant is 

h' n' u ', which then is the same as h'n'u (10-21) 

Since the invariant for the image in air must equal that for the image in the immersion 
lens, the two may be equated to yield 

h' = h/n' (10-22) 

This usage of the hemispherical surface corresponds to case b of Sec. 10.1.3.1. 

Design. To design an immersion lens, a diameter for the lens is selected larger than 
the unimmersed detector size and a few rays are traced from the primary optical system 
directed toward the edge of the field. The focus of these rays is examined to locate 
the optimum position for the detector. If the angles at which the rays strike the sur¬ 
face of the immersion element are large, the loss due to reflection may be too great. 
These angles may be reduced by increasing the radius of the immersion element. 
The radius used for an immersion element is usually a compromise among the desire 
for reasonable incidence angles, the space available for the element, and the rela¬ 
tively high costs of the material for immersion lenses. 

Immersion. To obtain the fullest gain from an immersion lens, the detector must 
be in optical contact with the plane surface. Some detectors can be deposited directly 
on this surface if the lens is made of a material compatible with the electrical and 
mechanical characteristics of the detector. If the detector index is higher than that 
of the immersion lens, the situation is ideal. If the detector index happens to be lower 
than that of the lens, or if there is a layer of air, or cement, or a low-index cover plate 
between the lens and the detector, then total internal reflection can occur at the inter¬ 
face where the light travels from high index to low. Since high-efficiency systems must 
operate with high convergence angles at the detector, the angles of incidence at these 
interfaces are often very large and T.I.R. is a problem unless the detector is truly 
immersed. 


434 


OPTICAL SYSTEMS 


Primary System (or Field Lens) 




Size of Detector 
Necessary without 
Immersion 


Fig. 10-16. Function of hemispherical immersion lens in reducing the 
size of the detector necessary to cover a given aperture or field of view. 


10.2.4.2. The "Aplanatic” Immersion Lens. Case c of paragraph 10 1.3.1. pro¬ 
vides another possible surface for use in an immersion lens, since the "aplanatic” 
surface may be inserted into a convergent beam without introducing aberration, just 
like the concentric surface of the hemispherical lens. As shown in Fig. 10-17, the 
use of this type of lens is limited to low-aperture systems. 

10.3. Refracting Objectives 

10.3.1. The Single-Element Objective. A single refracting element usually forms 
a very imperfect image. Chromatic aberration spreads out the image, and unless 
expensive aspheric surfaces are used, there is spherical aberration. Because the image 
quality required of many infrared systems is relatively poor, the simplicity and economy 
of a single spherical lens make it useful. 

The image quality of a single element can be estimated from the thin-lens contri¬ 
bution equations of Sec. 9.6.3. (Figures 10-35 through 10-41, except Fig. 10-36, allow 













REFRACTING OBJECTIVES 


435 



Fig. 10-17. Construction of "aplanatic” immersion lens showing how 
the acceptance angle is severely limited even for a zero angle field of 
view system, shown for n = 3.5 and an fl0.8 beam. 

estimation of the angular blur produced by a single element.) It is possible to deter¬ 
mine the size of the blur spot due to spherical aberration from Fig. 10-18. Each line 
on the plot represents an index of refraction. The x axis is the shape of the element, 
and the y axis is the size of the angular blur spot. To determine the spherical blur, 
one locates the proper refractive index line and finds the point on the line correspond¬ 
ing to the shape factor ( K ) of the lens. The shape factor is given by 

K = d/(ci - c 2 ) (10-23) 

where Ci and c 2 = the curvatures (or reciprocal radii) of the surfaces of the lenses. 
The blur factor is then read from the y axis. To convert this to angular blur in radians, 
multiply by y 3 <f> 3 , where y is the semi-aperture of the lens and </> is the power or recip¬ 
rocal focal length. The chart is derived from the equation for the third-order spherical 
aberration of a thin lens presented in Sec. 9.6.2, and applies to lenses with the object 
at infinity. The chart can be used for finite conjugates by splitting the lens into two 
portions (each of which is working at infinity) and adding the angular blur of both 
parts to get the final blur. 

To estimate chromatic blur, the index of refraction of the lens material is deter¬ 
mined for the extreme wavelengths of the spectral range over which the system is to 
be used. Divide their difference (n v — n r ) into the median index minus one to get 
the V value: 


V = (ti — 1 )/(n v — n r ) 


( 10 - 24 ) 












436 


OPTICAL SYSTEMS 



Fig. 10-18. Angular blur of simple lens as function of shape 
for various indices of refraction. 


Then for a lens at infinity the longitudinal chromatic aberration is 


LchC = flv (10-25) 

and the angular chromatic blur is 

_ , LchC 1 

/3C “ 2(f/no)f— 2V(/7no) (10-26) 

10.3.2. Achromats. Refer to Sec. 9.8. 

10.3.3. Combinations of Several Thin Single Elements. The use of a series of 
thin elements as shown in Fig. 10-19 makes it possible to reduce the spherical aber¬ 
ration of a refracting lens, although the chromatic aberration remains the same as 
for a single element. 





































REFLECTING OBJECTIVES 


437 



Fig. 10-19. Series of thin spherical lenses, each shaped 
to minimize spherical aberration. At an index of 1.75 
this three-element system can be designed to be com¬ 
pletely free of spherical aberration. 


If a set of elements with object at infinity is to have a total power </>, and each element 
of the set is bent to the shape for minimum spherical aberration, then the longitudinal 
spherical aberration of the set is given by 

j = i 

SC = 2 SCj (10-27) 

j=i 


SCj = 


— Y 2 (f)jn [4 n — 1 - 4j(J — 1 )(ra — l) 2 ] 
8 iHn - 1) 2 U + 2) 


where i = the number of elements in the set 
Y = the ray height at all elements 
<pi = the power of each element 
n = the index of the elements 
j = the element number 


(10-28) 


Equation (10-28) has been worked out for i = 1, 2, 3, and 4 and is presented in Fig. 
10-20. To use the graph, the number of elements i and the index n are determined. 
The intersection of the i line and the n coordinate extended to the y axis indicates 
either the longitudinal spherical or the angular diameter of the blur spot. 

To fabricate such a system the radii must then be selected so that each element is 
bent to minimize its spherical aberration contribution. This can be done with Eq. 
(9-215) and also Eq. (9-203), (9-204), (9-205), (9-206), and (9-226). 

It is practical to use these systems at finite conjugates by arranging two such systems 
(of appropriate focal lengths) facing each other with parallel light between them, as 
shown in Fig. 10-21. 

10.4. Reflecting Objectives 

The pure reflecting objective is completely free of chromatic aberration and has 
no transmission losses. A reflector surface is usually coated with a thin evaporated 
aluminum film, which has an excellent infrared reflectance and which in itself is dura¬ 
ble. Other materials are occasionally used for reflecting films. 

10.4.1. The Spherical Reflector. The simple spherical reflector is a useful infrared 
device. It is simple, inexpensive, and easy to align and mount. When used with an 
aperture stop at its center of curvature, it has only spherical aberration. Figure 
10-22 illustrates this usage of a spherical mirror. When the stop is at the center of 







LONGITUDINAL SPHERICAL ABERRATION, y <t> 


438 


OPTICAL SYSTEMS 



Fig. 10-20. Thin lens aberration of i thin elements of index n, each element 
bent for minimum spherical aberration. 



Fig. 10-21. Minimum spherical elements as relay system. 

For smaller fields, the image quality of this type of system is 
limited only by chromatic aberration, since spherical aber¬ 
ration is eliminated and coma is very small. 

curvature, any principal ray (through the center of the stop) may be considered an axis 
of the system. Thus the image quality for any field of view is almost the same as the 
on-axis image quality. The image surface is a sphere with a radius approximately 
one-half of the mirror radius and concentric with the mirror. 

The aberrations of a spherical mirror can be estimated by the third-order surface- 
contribution equations of Sec. 9.6.2. By setting n — 1.0 and n' = —1.0, the following 
expressions can be obtained for the third-order aberrations of a sphere with object at 
infinity: 

Spherical: 

SC = y 2 !4r (10-29) 


Sagittal coma: 


CC* = y 2 (l P 


r)u p 2 l2r 2 


(10-30) 


DIAMETER OF BLUR SPOT, yV 




















REFLECTING OBJECTIVES 


439 



Fig. 10-22. Spherical reflector with stop at center of 
curvature. In this arrangement the mirror has no coma 
or astigmatism and the focal surface is a sphere of radius 
r/2 about the center of curvature. Only spherical aber¬ 
ration limits the image quality. 


Astigmatism: 

AC*= (l p — r) 2 u p 2 /4r 

Petzval curvature: 

PC = Up 2 rlA 

where y = the height of the axial ray from the axis 

r — the radius of the sphere 

lp = the distance from sphere to the stop 

Up = the slope of the principal ray 

* = new third-order coefficient 

Note that, if the stop is placed at the center of curvature, r=l p , AC*= 0. and CC* = 0. 
If the stop is placed at the mirror, however, l p = 0 and Eq. (10-30) and (10-31) reduce to 

Sagittal coma: 

CC = —y 2 Up/2r 

Astigmatism: 

AC = —ru p 2 IA 

The sagittal and tangential field curvature can be obtained from 

X, = PC + AC = ~~ + (y — 2 ) 


(10-33) 

(10-34) 

(10-35) 


(10-31) 

(10-32) 


u p 2 r 

2 


if l p = 0 


X, = PC 4- 3AC = u p 2 r ^UpHp^j - 2^ 


(10-36) 


= u p 2 r if l p = 0 















440 


OPTICAL SYSTEMS 


Where the sphere is used at finite conjugates the following expression is useful: 

SC = (m — l) 2 y 2 /4r (10-37) 

where m is the magnification, 


h' 



i u 
l u' 


10.4.2. The Parabolic Reflector. For distant objects located on the optical axis, 
a paraboloid is completely free of aberrations and its image quality is diffraction lim¬ 
ited. The axial image of a perfectly made parabola is an airy disc as described in Sec. 
9.9. The paraboloid is used in many infrared systems requiring good image quality 
over a very small field. 

Off axis the paraboloid has the same amount of coma and astigmatism as a spherical 
mirror when the stop is at the mirror (in both cases). The stop shift theory of Sec. 
9.6.3 indicates that, because there is no spherical aberration, coma will not change 
as the stop is moved. Thus for all stop positions the coma of a paraboloid is given by 


CC* = y 2 u p /4f 


(10-38) 


Since a paraboloid has coma, the astigmatism will vary with stop position. The astig¬ 
matism with the stop at the mirror is given by 

AC = — fu p 2 /2 (10-39) 

and drops to zero when the stop is at the focal plane position Up = f ). The Petzval sur¬ 
face is the same as that of a sphere. Thus with the aperture at the focal plane, both s 
and t foci lie on a spherical surface of radius f. 

A high-quality paraboloid may be an order of magnitude more costly than that of 
an equivalent spherical surface, which is often of much better optical accuracy. Deep 
paraboloids are especially difficult to fabricate precisely. Before specifying a parab¬ 
oloid for a system, one must investigate whether a spherical surface will meet the 
requirements of the system. 

The optical collimator produces a source of radiation which appears to be at infinity, 
so that rays from any given point of the source are parallel to each other. The prime 
purpose for most collimators is the testing of other optical systems, and thus their 
optical quality must be superior to the device being tested. Since a paraboloid has 
no aberrations (when used on axis), it is an ideal objective for use in an infrared col¬ 
limator. To avoid obscuration of the central portion of the collimated beam by the 
light source, the aperture of the collimator is often decentered from the paraboloid 
axis, resulting in the so-called off-axis paraboloid as illustrated in Fig. 10-23. 



Fig. 10-23. Typical infrared collimator with paraboloid 
reflector objective, showing use of off-axis aperture. 













REFLECTING OBJECTIVES 


441 


10.4.3. Other Conics: The Ellipsoid and Hyperboloid. In the ellipsoid and the 
hyperboloid, rays directed toward, or emerging from, one focus, are reflected toward 
the other without spherical aberration. Thus the sphere has perfect imagery of points 
at its center, which is the locus of both foci. At the other extreme, the paraboloid 
has perfect imagery between a focus at infinity and the point usually thought of as its 
focus. The ellipsoid and the hyperboloid image from one focus to the other. Both, 
however, have large amounts of coma so that they cannot be used to form good images 
of extended objects. 

The fabrication costs discussed in Sec. 10.4.2. are even more applicable to the ellip¬ 
soid and hyperboloid, as is the advisability of considering the substitution of a spherical 
surface whenever possible. 

10.4.4. Compound, or Double, Reflecting Systems. In many applications the 
location of the image plane in the path of the incoming rays is inconvenient. Thus 
a great number of objective systems have been devised to get around this difficulty by 
inserting a second mirror to place the focus to one side of the incoming radiation or 
behind the primary mirror. 

10.4.4.1. The Newtonian Telescope. The basic system used for most small astronom¬ 
ical reflecting telescopes is a plane mirror located near the focus of the primary mirror 
(conventionally a paraboloid) and at 45° to the optical axis so that the focus is located 
immediately outside the incoming beam, as shown in Fig. 10-24. 



10.4.4.2. The Folded Reflector. The system shown in Fig. 10-25 is the simplest 
way of locating the focal point behind the primary. A plane mirror, perpendicular 
to the axis and lying between the primary and its image reverses the direction of the 
light, so that it passes through the central hole cut in the primary. This places the focal 
point beyond the mirror where it is more accessible. However, this mirror obscures 
at least half the aperture. 



Fig. 10-25. Folded reflector. 











442 OPTICAL SYSTEMS 

10.4.4.3. The Cassegrainian Objective. If the secondary mirror of Fig. 10-25 is made 
convex, the focal length of the objective is increased, and the obscuration of the beam 
by the secondary mirror is reduced. Figure 10-26 shows a typical Cassegrain. 

The original Cassegrainian objective had a paraboloid for the primary and a hyper¬ 
boloid for the secondary. Both were used in such a way that the image formed by each 
was free of spherical aberration (see Sec. 10.4.3). The Cassegrain is widely used in 
infrared work because of the accessibility of its focal point and its relatively low obscu¬ 
ration ratio. By proper selection of "radii,” the Petzval surface can be made flat. 


Convex Secondary Concave Primary 



10.4.4.4. The Gregorian Telescope. The original Gregorian objective consisted of 
parabolic primary mirror and an elliptical secondary which was placed beyond the 
focus of the primary mirror so that its "object” and image were located at the foci of 
the ellipse. Thus, there is a real internal image in the Gregorian, and the final image 
is erect, as shown in Fig. 10-27. 

The Gregorian is rarely used because it is a longer system than the Cassegrainian 
and offers no compensating advantages. 



10.4.4.5. Baffling of Folded Systems. All folded systems must be baffled to prevent 
extraneous radiation from flooding the focal plane. In systems which require a large 
field coverage and high relative aperture, complete baffling is complex, and in extreme 
cases may be nearly impossible. 

Figure 10-28(a) shows a typical Cassegrainian objective. The useful rays are drawn 
as solid lines. The dashed lines indicate the path of undesired radiation which passes 
through the system without striking either mirror and which can flood a focal plane 
detector with stray light. The placement of baffles and sunshades in such a system 
to cut off stray light is shown in Fig. 10-28(6). 









CATADIOPTRIC OBJECTIVES 


443 




Fig. 10-28. Folded systems (a) before baffling, (6) after baffling. 

A field lens that is carefully designed to image the secondary mirror (and nothing 
more) on the detector surface will also eliminate the stray radiation. However, its 
material may absorb (or reflect) a portion of the desired radiation, and secondary sur¬ 
face reflections of the stray radiation may still get to the detector. Where possible, 
the field lens plus baffles make a strong combination. 

10.5. Catadioptric Objectives 

A catadioptric objective combines reflecting and refracting elements. The systems 
which are most widely used in infrared work are those with relatively thin refracting 
components, because these absorb the least energy, and, in general, have the least 
chromatic aberration. 

10.5.1. The Schmidt System. The Schmidt objective combines the advantages 
of the sphere and the paraboloid. As shown in Fig. 10-29, it consists of an aspheric 
corrector plate at the center of curvature of a spherical mirror. The surface of the cor¬ 
rector is shaped to compensate for the spherical aberration of the reflector. The 
corrector is located at the center of curvature, as is the aperture stop, so that the system 
is free of coma and astigmatism as well. 

Aspheric surfaces most often chosen are those which have their minimum thickness 
at the 0.707 or 0.866 zone (of the marginal ray height) depending on whether it is 
desired to minimize chromatic aberration or optimize the off-axis correction. 

The Schmidt system does not achieve perfect imagery because the off-axis ray bundles 
do not strike the corrector at the same angle as the on-axis bundles. The effect produced 























444 


OPTICAL SYSTEMS 


Aspheric 


Spherical 

Reflector 



is to increase the effect of the corrector, producing an overcorrected condition off axis. 
(The effect of this on the angular resolution can be estimated from Fig. 10-39 and Eq. 
10-51.) The Schmidt performance may be improved somewhat by one or more of the 
following techniques: 

1. Incompletely correcting the axial image to reduce the overcorrection off axis. 

2. Using a slightly aspheric primary mirror to reduce the correction load on, and 
thus the overcorrection introduced by, the corrector. 

3. "Bending” the corrector slightly. 

4. Using more than one corrector. 

5. Using an achromatized corrector. 

The aspheric corrector of the Schmidt system is usually easier to fabricate than the 
aspheric surface of the paraboloid. This is true because the index difference across 
the corrector, which is usually glass, is only 0.5 compared to the effective index dif¬ 
ference of 2.0 across the reflecting surface of the paraboloid, making it only one-fourth 
as sensitive to fabrication errors. Linfoot [10] and Bouwers [11] have published con¬ 
siderable information about the design and performance of the Schmidt system. 

10.5.2. The Mangin Mirror. In the Mangin mirror, Fig. 10-30(a), the spherical 
aberration is corrected by the introduction of a negative lens element in contact with 
the reflector. For large relative apertures the marginal spherical aberration can be 
corrected, but a residual remains. A large penalty is paid in the form of the chromatic 
aberration introduced by the negative refracting element. The Mangin mirror is, 
however, relatively inexpensive to fabricate and simple to mount. 

Figure 10-30( b) shows a system with a secondary Mangin mirror, in a sort of Cas- 
segrainian arrangement. This can be used with a Mangin primary to reduce the 
residual spherical aberration, or with a spherical primary to reduce the cost and weight 
of the system. The secondary may be designed to be the "power equivalent” of a con¬ 
vex, concave, or plane mirror. 

The Mangin mirror may be achromatized by making the negative element an achro¬ 
matic doublet. Section 10.6 indicates the image quality of the basic Mangin mirror. 













CATADIOPTRIC OBJECTIVES 


445 


Second Surface 
Mirror 




10.5.3. The Bouwers (Maksutov) Objective [11]. The principle of correcting 
the aberration of a spherical mirror with a negative power-refracting element is fur¬ 
ther refined in the Bouwers system. By moving the corrector away from the mirror, 
it is possible to use two degrees of freedom (shape and position) to make a radical 
improvement over the Mangin mirror. 

Figure 10-31 illustrates the concentric system of Bouwers, in which the corrector 
is placed well away from the mirror and all radii are concentric. This system has 
the same advantages as the simple sphere with its stop at the center of curvature, 
in that the image quality is the same for all field angles. The residual spherical 
aberration of this type of system is low, as is the chromatic aberration, so that an ex¬ 
cellent objective results. The design of such a system is easy and fast because there 


Aperture Stop 



Fig. 10-31. Bouwers concentric system. 










446 


OPTICAL SYSTEMS 


are only three variables: the three radii. The technique is to fix n at an arbitrary 
value; then, for each value chosen for r-z, a value for can be found for which the mar¬ 
ginal spherical aberration is zero. Thus the entire range of designs can be surveyed 
quickly, because only a few rays need to be traced as all points of the field have the 
same image quality. 

The concentric principle allows the corrector to work identically in either of two 
positions, behind or before the common center of the system as indicated in Fig. 10-31 
by the dashed lines. In its alternate position in front of the common center, the optical 
effects of the correction are exactly the same; however, this position is used when an 
extremely high state of correction is desired in order to avoid interference between 
corrector and focal plane. 

The corrector may also be used as the support for a Cassegrainian secondary, as 
shown in Fig. 10-32 in several different arrangements. 



Fig. 10-32. Some arrangements of Bouwers systems as Cas¬ 
segrainian objectives. An arrangement similar to (c) is often 
used in missile guidance since the corrector makes a rea¬ 
sonably aerodynamic window or dome. 

The basic Bouwers system has residuals of chromatic and spherical aberration, 
which although small are often worth correcting. The chromatic aberration may be 
reduced by making the corrector a two-element component, with materials and the 
"buried” surface chosen to achromatize the corrector, as shown in Fig. 10-33. The 
residual spherical aberration may be eliminated by the use of a Schmidt type aspheric 
corrector, located at the common center of the system. Since the power of this aspheric 
corrector is much less than that of a Schmidt, its correction variation with obliquity 
is correspondingly less, and an excellent system results. 

A number of variations on these principles have been produced, using dual correctors 
(in front and behind center), multiple aspherics, and achromatized aspherics, each of 
which is designed to improve still further on this excellent system. 












CATADIOPTRIC OBJECTIVES 


447 



Fig. 10-33. Bouwers system with 
achromatized corrector plate. 


One reason for the popularity of the basic Bouwers system, over and above its ex¬ 
cellent performance and easy design, is that, being composed entirely of spherical 
surfaces, it is relatively inexpensive to make. 

The arrangement of the Bouwers with the corrector convex to the incident radiation 
is widely used, often with a Cassegrainian secondary of one sort or another, for infra¬ 
red missile guidance systems. The "dome” formed by the corrector then functions as 
a window for the system as well as a corrector. The corrector need not be concentric 
when a large well-corrected field is not required. See Fig. 10-34. 


Concentric 

Corrector 



Fig. 10-34. Corrected concentric Bouwers system. 


10.5.4. Aberrations of a Plane Parallel Plate in a Convergent Beam. It is often 
necessary to insert a filter or beam splitter into an optical system in a location at 
which the beam is not parallel, usually to minimize the size of the insert. This plane 
parallel plate in a convergent beam introduces aberrations that may significantly 
affect the performance of the system. The following expressions can be used to evaluate 
the aberrations introduced by this plate. 

Spherical Aberration: 


Third Order: 


SC = 


tu 2 {n 2 — 1) 
2N 3 


(10-40) 












448 


OPTICAL SYSTEMS 


Trigonometric: 



n cos u 


Vrc 2 — sin 2 u , 


(10-41) 


Astigmatism: 
Third Order: 

Coddington: 

/'_/' = 
L s * t 


AC = 


tu p 2 (n 2 — 1) 


t 


n 2 cos 2 u p 


- 1 


Vrc 2 — sin 2 Up 
Coma-Third Order: 

CC = 


_(N 2 — sin 2 Up) 
tu 2 u p {n 2 — 1) 


= 0.211 at u , 


2 n 3 


Longitudinal Chromatic-Third Order: 

LchC = 


lAiV _ ~ 1) 

n 2 n 2 V 

where t = the thickness of the plate 

n = the index of refraction of the plate 
u = the convergence angle of the rays to the axis 
u p = the tilt of the plate from normal to the axis 
V = (n — 1)/A n — Abbe V number. 


(10-42) 

45 ° ( 10 - 43 ) 


(10-44) 

(10-45) 


These equations are derived directly from the surface-contribution equations and the 
ray-tracing equations of Chapter 9. 

10.6. Rapid Estimation of Image Blur Size for Several Optical Systems 

The performance required of an optical system usually can be expressed in terms 
of resolution or image spot size, that is, the size of the smallest blur of light that the 
system is capable of producing as the image of an infinitesimally small source of light. 
The minimum size of the image spot is determined either by the size of the system 
aperture (because of the wave nature of light) or by design characteristics (that is, 
aberrations) of the optical system. The accompanying figures provide a convenient 
tool to determine the limiting size of the blur spot for diffraction-limited systems and 
for several specific optical systems which are widely used. 

The figures express the image spot size in one of two ways: as the linear diameter 
(B) of the image spot, and as the angular diameter (/3). The angular diameter /3 
is simply the linear diameter B divided by the effective focal length of the optical 
system. 

10.6.1. Diffraction-Limited Systems. The image size of any optical system is 
always limited by the wavelength of the radiation involved. In a perfect system 
(that is, one which has no aberrations or defects of manufacture) this wavelength 
limitation will determine the minimum size of the blur spot image. The Airy disc, 
as this blur is called, takes the form of a central blur of light surrounded by alternating 
light and dark rings of rapidly decreasing intensity. (See also Sec. 9.9.) The central 
blur contains 84% of the energy, so that we can consider the diameter of the first dark 
ring about this disc as a conservative value of the effective size of the blur spot. The 
diameter of this dark ring is given by the expression 

B = 2.44\(/7no) 


(10-46) 













449 


RAPID ESTIMATION OF IMAGE BLUR SIZE 
and the angular size of the blur is given by 

/3 = 2.44ALU 1 rad (10-47) 

where \ = the wavelength of the radiation 

f! no = the relative aperture or speed of the system (that is, the ratio of focal length 
to effective diameter) 

D = the effective diameter of the system 

Figure 10-35 provides a rapid method of evaluating these expressions for the optical 
systems shown in Fig. 10-36. Paralleling the x and y axes of the chart are a series of 
spaced lines each labeled with a wavelength from 0.5 to 32 p. At an angle to the 
wavelength lines is another set of lines. The x-axis set represents the effective optical 
diameter in inches; the y-axis set represents the relative aperture. The intersection 
of an angled line with the appropriate wavelength line indicates the size of the min¬ 
imum blur for a given system. 

Example 1 

An optical system with a focal length of 10 in. and a clear aperture of 5 in. working 
at a wavelength of 2 /x is required. The intersection of the 5-in.-diameter line with 
the 2-/z wavelength line falls on the abscissa corresponding to an angular blur of 

0 , ANGULAR BLUR SPOT (rad) 


10 10 10 10 



0 , ANGULAR BLUR SPOT (rad) 

Fig. 10-35. Blur spot size chart for diffraction-limited systems. 











































































450 


OPTICAL SYSTEMS 







(e) "Rear" Concentric (f) "Front" Concentric 

Fig. 10-36. Schematic sections of some 
common optical systems. 


3.8 ± 10~ 5 rad. The speed (relative aperture) of this system is /72 (10-in. focal length 
divided by the 5-in. diameter). The intersection of the /72 line with the 2-/x line 
falls on the ordinate corresponding to 3.8 X 10 4 in. 

Note that a blur spot diameter of 3.8 x 10 -4 in. subtends an angle of 3.8 x 10~ 5 
rad from a distance (focal length) of 10 in. The diagonal focal-length lines of the 
chart can be used to convert from linear blur diameter to angular blur diameter 
(and vice versa) by following a diameter line across to the intersection with the 
proper focal length line and then noting the angular size corresponding to the inter¬ 
section. 

Real optical systems usually produce blur spot images which are larger than the dif¬ 
fraction limit described above. Larger blur spots may result from aberrations, that 
is, the failure of the optical system to produce a perfect focus. In many systems, the 
size of the aberration blur spot can be expressed as a function of relative aperture 
(ft no) and field of view (degrees off axis). The following material represents formulas 
and figures which permit rapid determination of the aberration blur characteristics 
of several widely used systems. 

10.6.2. Spherical Reflector. A simple spherical reflector (Fig. 10-36a) with its 
limiting aperture at the center of curvature has only one image defect, spherical aber¬ 
ration, which is constant over the entire field of view. The minimum angular size 
of the blur spot is given by 


/3 = 7.8 X 10~ 3 X (/7no) -3 rad 


(10-48) 
















RAPID ESTIMATION OF IMAGE BLUR SIZE 


451 


0, ANGULAR BLUR SPOT (rad) 



for modest apertures (for large apertures the constant is somewhat larger, for example, 
9.1 X 10 -3 for f/1). To evaluate this expression, locate the desired speed (Fig. 10-37CA) 
and read the corresponding angular blur (/ 3) from the abscissa scale. 

When the limiting aperture is not at the center of curvature, coma and astigmatism 
are present in the off-axis image. In a typical coma blur patch, most of the energy is 
contained in a small triangular area which is only one-third of the total height of the 
full coma patch. (See Fig. 9-11 for a sketch of a typical coma patch.) The size of the 
small triangular (sagittal) coma patch (coma*) produced by a spherical reflector when 
the aperture stop is at the surface of the reflector is given by 

Coma* /3 = 0.0625 x d(fl no) -2 rad (10-49) 

where 6 is the angle (in radians) that the object and image are off axis. To use Fig. 
10-37(5) to determine sagittal coma, select the horizontal line representing the angle 
6 and locate the intersection with the relative aperture line; then read up to find /3. 

The minimum blur produced by the astigmatism with the stop at the reflector is given 
by 

astig. (3 = 0.5 X 0 2 x (/7no) _1 rad (10-50) 

and Fig. 10-37(0 may be used to evaluate this expression in the same manner as (5). 
Example 2 

Assume a spherical reflector (stop at the reflector) with a 10-in. focal length, 
5-in. clear aperture (relative aperture /V2) with a field of ± 10° (total field 20°). From 
Fig. 10-37, the angular blur due to 

spherical aberration will be 1.0 X 10 -3 rad 

sagittal coma at 10° will be 2.7 X 10 -3 rad 

astigmatism at 10° will be 7.6 X 10 -3 rad 













































































452 


OPTICAL SYSTEMS 


The interaction of the various aberrations is, in general, difficult to predict precisely. 
For most purposes, however, it is sufficient to simply add the blurs and accept the sum 
as a good indication of the final image blur size. Thus, for this example, the indicated 
angular blur at the edge of the field would be about 11 mrad (11 x 10~ 3 rad), about 
ten times as large as that at the center of the field. 

If the aperture stop were between the center of curvature and the mirror, the coma 
and astigmatism blurs could be approximated by interpolation between the zero value 
obtained with the stop at the center and the values given above. Coma is a linear func¬ 
tion of the separation of stop and center of curvature; astigmatism is a function of 
the square of this separtion. 

10.6.3. Mangin Mirror. The Mangin mirror (Fig. 10-366) is a second surface 
reflector whose refracting first surface is used to correct the spherical aberration of 
the reflecting surface. The refraction introduces chromatic aberration, however, 
and limits the improvement attainable by this technique. Figure 10-38 indicates the 
angular blur of a Mangin mirror due to residual spherical aberration and chromatic 
aberration. The sagittal coma blur of the Mangin is about half that of a similar 
spherical mirror. 




llfltl 1 t 

♦i 11 i*i i * r*' ii i ♦tit* rr~ 

.57 .6 

Blur 

• 8 .9 1 1.1 1 

due to Spherical Aberi 

.2 1.4 1.6 2 

ration 

Mill H 

mm^: 

ini 11 i i 

.6 .8 1 1.4 2 2.8 4 

1 

Blur due to Chromatic Aber 
(V = 100) 

l . 

ration 


10' 2 10' 3 10' 4 10 


/3, ANGULAR BLUR SPOT (rad) 

Fig. 10-38. Blur spot size chart for Mangin mirrors. 


-5 


Example 3 

Figure 10-38 indicates that an /72 Mangin mirror has a spherical aberration blur 
of 1.5 x 10~ 4 rad and a chromatic blur of about 8 X 10 -4 rad, if the material used 
has a V value of 100 (see Sec. 10.6.7 on refracting systems for definition of V value). 
If the system operated in the near infrared (1 to 2.3 p), the equivalent V value of 
glass for this region would be about 35 and the chromatic blur spot would be 100/35 
times as large, that is, about 2.3 X 10~ 3 rad. The sagittal coma blur would be about 
1.4 X 10 -3 rad at 10° off axis, half that indicated in Example 2. 

10.6.4. Parabolic Reflector. The parabolic reflector, or paraboloid, is widely 
used because of its perfect image quality on axis, where the angular blur size is theoret¬ 
ically limited only by diffraction as discussed above in Example 1. For off-axis usage, 
however, coma and astigmatism are present in the same amounts as in a spherical 
reflector when the stop is in contact with the mirror. (Note that the coma of a para¬ 
bolic reflector is constant and not a function of stop position, but the astigmatism is 
zero when the stop is located at the focal plane, that is, one focal length before the 
mirror.) Thus the angular blur size for a parabolic reflector can be determined from 
the same equations (10-49 and 10-50), and the same chart (Fig. 10-37) used for a spheri¬ 
cal reflector. If we substitute a paraboloid of the same dimensions in Example 2, 
the blur on axis would be that of Example 1, or 3.8 X 10 -5 rad at a wavelength of 2 /z, 











RAPID ESTIMATION OF IMAGE BLUR SIZE 453 

and the off-axis blurs caused by coma and astigmatism would be 2.7 x 10~ 3 and 7.6 
X 10 -3 rad, respectively, as in Example 2. 

10.6.5. Schmidt System. The Schmidt system (Fig. 10-36c) combines the perfect 
axial image quality of the paraboloid with the uniform image quality over a wide 
field. This is accomplished by the use of a spherical primary mirror with a thin re¬ 
fracting aspheric corrector plate located at the center of curvature of the primary. 
Thus, the Schmidt system is diffraction limited on axis; the size of the angular blur 
spot off axis is given by 

/8 = 0.0417 x x (/'/no) -3 (10-51) 

A Schmidt system with an ft 2 relative aperture would have a 1.6 X 10 -4 -rad blur 
at 10° off axis. Figure 10-39 is used in the same manner as Fig. 10-37(5) and (C). 



0, ANGULAR BLUR SPOT (rad) 

Fig. 10-39. Blur spot size chart for Schmidt systems. 


10.6.6. Catadioptric Systems. Systems which combine refracting and reflecting 
elements are known as catadioptric systems (Fig. 10-36d, e, and f). Perhaps the 
most useful of these is the concentric Bouwers system. Because of its concentric 
character the image blur size is the same over the entire field of view. Figure 10-40 
presents the limiting characteristics for two forms of the concentric Bouwers and also 
for the corrected concentric Bouwers, which is a concentric Bouwers with a Schmidt- 
type corrector at the center of curvature. 



0, ANGULAR BLUR SPOT (rad) 

Fig. 10-40. Blur spot size chart for concentric Bouwers systems. 


















































































454 


OPTICAL SYSTEMS 


10.6.7. Refracting Systems. Because of the complexity of refracting systems, 
only the most elementary types lend themselves to anything approaching a complete 
analysis. The paragraphs below consider only a single thin element with the aperture 
stop at the element. 

The spherical aberration of a thin lens is a function of its index of refraction (a char¬ 
acteristic of its material) and its shape. Figure 10-41CA) indicates the angular blur 
due to spherical aberration of a lens shaped to minimize aberration, when the surfaces 
of the lens are spherical. (Spherical aberration can be eliminated by the use of aspheric 
surfaces.) The example of an fl 2 optical system as a lens would thus have a blur of 
8 x 10- 3 rad if made of glass (n = 1.5) or 1 x 10 ~ 3 if made of germanium (n = 4). 



v = 10 

V = 20 

V = 40 

V = 80 

V = 160 

V = 320 


Chart A Blur Due to Spherical Aberration 
at "minimum bending") 



100%— f /* 

f/2 

f/4 

f/8 

f/16 

f/32 

f/64 

f/12 8 

75-90%— 

f/1 

f/2 

f/4 

f/8 

f/16 

f/32 

f/64 

40-60% — 


f/1 

f/2 

f/4 

f/8 

f/16 

f/32 


10 


-2 


Chart B Blur Due to Chromatic Aberration 

I I 

10‘ 3 uf 4 

/ 3 , ANGULAR BLUR SPOT (rad) 


V = 10 

V = 20 

V = 40 

V = 80 

V = 160 

V = 320 


10 


-5 


Fig. 10-41. Blur spot size chart for single refracting element. 


The coma blur of a thin lens, shaped for minimum spherical aberration, with the stop 
in contact with the lens, is given by 


coma s B 


_ 0 _ 

16U + 2)(/7no) 2 


The coma is zero if the stop position is given by 


(10-52) 


l pr = 2(n - 1) 2 /<M1 - 4n) (10-53) 

The astigmatism of a lens in contact with the aperture is given by Eq. (10-53) so that, 
for lenses of "minimum bending,” the as given by Fig. 10-37(0 can be used as an in¬ 
dication of the off-axis image characteristics, since the coma of a minimum bending 
lens is usually quite small. 

A simple lens also has chromatic aberration, which is the difference in focal position 
for light of various wavelengths. This results from a variation of the refractive index 
of the material of the lens with wavelength. A sort of chromatic "figure of merit” 
for optical materials is the Abbe V value, which when generalized for use at wave¬ 
lengths other than the visual, is expressed as 


V = (n m — 1) X (n s — ni ) _1 


(10-54) 































































RAPID ESTIMATION OF IMAGE BLUR SIZE 


455 


where n is the refractive index and the subscripts m, s, and / refer to the index at the 
middle, short, and long wavelengths of the spectral band of sensitivity of the detecting 
device used with the optical system. Suppliers’ catalogs should be consulted for exact 
index values, from which V may be computed. For optical glass some t5 r pical values 
are: 


Visible spectrum V = 30 to 65 

Lead sulfide region 
0.4 to 2.5 n V = 9 to 15 

Near infrared 

1 to 2.3 /x V = 30 to 40 

The angular blur due to chromatic aberration is given by 

chromatic /3 = 0.5V~ 1 (f/no)~ 1 (10-55) 

and can be found by use of Fig. 10-37(B) when V and ft no are known. Note that the 
first row of /'/no values indicates the diameter of the blur containing 100% of the energy. 
The second row indicates the blur containing 75 to 90% of the energy, and the third 
row the blur containing 40 to 60% of the total energy. The ordinate corresponding 
to the intersection of the V value lines and the appropriate fl no line indicates the 
value of the angular blur spot (3. 

10.6.8. Summary of Equations. 

Diffraction limit 



/3 = 2.44 XXX D -1 

(10-56) 

Spherical reflector 



Spherical aberration 

13 = 0.0078 (//no)' 3 

(10-57) 

Sagittal coma 

(3 = 0.0625 x 6 x (/'/no) -2 

(10-58) 

Astigmatism 

• 



/3= 0.5 x d 2 x (//no)- 1 

(10-59) 

Coma and astigmatism are 

zero when stop is at the center of curvature. 

Coma is a 


linear and astigmatism is a square function of stop to center-of-curvature distance. 
Parabolic reflector 

Astigmatism and coma blurs per Eq. (10-58) and (10-59). 

Coma does not vary with stop position. 

Astigmatism is zero for stop at focal plane. 

Mangin Mirror 

Figure 10-38 gives computed spherical and chromatic blurs. 

Coma blur is approximately one-half that of Eq. (10-58). 

Schmidt System 
Total meridional blur 


/3 = O.O4170 2 x (f/no)~ 3 


(10-60) 


Bouwers systems 

Figure 10-38 gives computed blurs. 


456 


OPTICAL SYSTEMS 


Single-element refracting (spherical surfaces) 
Spherical 


Coma 


Chromatic 


(3 = K(f/no)~ 3 

K-f(n) = 0.067 for n = 1.5 
0.027 for n = 2 
0.0129 for n = 3 
0.0087 for n = 4 


/3 = 6/[16(n + 2) (/’/no) 2 ] 
(3 = 0.5V-' (/Vno)- 1 


(10-61) 


(10-62) 

(10-63) 


Astigmatism per Eq. (10-59). 

References 

1. A. C. Hardy and F. H. Perrin, Principles of Optics (McGraw-Hill, New York (1932)). 

2. P. Drude, The Theory of Optics Dover, Publication, Inc., New York (1959). 

3. M. Born and E. Wolf, Principles of Optics, Pergamon, New York (1959) First ed., 148-149. 

4. R. C. Jones, Proc. IRIS, 6, 4, 1 (1961). 

5. W. L. Wolfe and J. Duncan, Proc. IRIS, 6, 2, 25-28 (1961). 

6. R. DeWaard and E. Wormser, Thermistor Infrared Detectors, Part I, Properties and Develop¬ 
ments, NAVORD 5495, Barnes Engineering Co., Stamford, Conn. (1958). 

7. A. E. Murray, Proc. IRIS, 6, 1, 145 (1961). 

8. D. E. Williamson, J. Opt. Soc. Am., 42, 712 (1952). 

9. Optical Light Pipe Design Study, Final Report, Contract No. HAC P. O. 4-541998-FF-90-3, 
Santa Barbara Research Center, Goleta, Calif. (1960). 

10. E. H. Linfoot, Recent Advances in Optics, Oxford University Press, Amen House, London, 
E.C. 4(1955). 

11. A. Bouwers, Achievements in Optics (Elsevier, Netherlands (1950)). 

Bibliography 

Design Examples of Cassegrain and Schmidt Type Optics, 5, Institute of Optics, The University of 
Rochester, Rochester, New York (1963). 

Geometrical Optics, I, Institute of Optics, The University of Rochester, Rochester, New York (1963). 
Image Evaluation Techniques, 3, Institute of Optics, The University of Rochester, Rochester, New 
York (1963). 

Optical Design Techniques, 2, Institute of Optics, The University of Rochester, Rochester, New 
York (1963). 


Chapter 11 
DETECTORS 


T. Limperis 

The University of Michigan 


CONTENTS 


11.1. Introduction. 458 

11.1.1. Responsive Elements. 458 

11.1.2. Windows. 459 

11.1.3. Apertures. 459 

11.1.4. Dewar Flask. 459 

11.2. Detector Parameters. 459 

11.3. Data Enumeration. 460 

11.4. Test Procedures. 501 

11.4.1. Determination of NEP. 501 

11.4.2. Time Constant. 504 

11.4.3. Frequency Response. 504 

11.4.4. Pulse Response. 505 

11.4.5. Spectral Response. 508 

11.4.6. Noise Spectrum. 510 

11.4.7. Sensitivity Contours. 510 

11.4.8. General Comments. 512 

11.5. Theoretical Limit of Detectivity. 512 

11.5.1. Derivation of D* for Photon Noise Limitation. 513 

11.5.2. System Design Considerations. 515 


457 























11. Detectors 


11.1. Introduction 

A detector is a device providing an electrical output that is a useful measure of inci¬ 
dent radiation. It includes not only the responsive element but also the windows, 
limiting aperture and dewar flask. 

11.1.1. Responsive Elements. The responsive element is a radiation transducer. 
It changes the incoming radiant power into an electrical power which can be amplified 
by the accompanying electronics. The transduction methods can be separated into 
two groups. In the first group (thermal detectors), the responsive element is sensitive 
to changes in its temperature, brought about by fluctuations of the incident radiation 
power. The second group (photodetectors) contains responsive elements which are 
sensitive to fluctuations in the number of incident photons. The detectors described 
in this chapter are listed below. A short description of each detection process follows 
the listing. 


Group 1 Thermal Detectors 

Thermistors (bolometric) 
Thermocouples (thermovoltaic) 
Thermopiles (thermovoltaic) 
Golay cells (thermopneumatic) 


Group 2 Photodetectors 


GaAs 


Cu-Cu 2 0 
In As 
InSb 


Photovoltaic 


Si 

PbS 

PbSe 

PbTe 

Te 

InSb 

Ge:Au ► 

Ge:Cu 

Ge:Hg 

Ge:Cd 

Ge:Zn 

Ge-Si:Zn 

Ge-SiiAu, 


Photoconductive 


InSb ) Photoelectro- 

HgTe (5% ZnTe, 5% CdTe) 1 magnetic 

Various mixtures of ) ™ , . . 

n ,. . . ) Photoemissive 

alkali earth oxides J 


458 




DETECTOR PARAMETERS 


459 


Bolometric: Changes in temperature of the responsive element induced by the 
incident radiation causes a change in the electrical conductivity of the element. This 
change in conductivity is monitored electrically. 

Thermovoltaic: When the temperature of a junction of dissimilar metals fluctuates 
because of changes in the level of incident radiation, the voltage generated by the 
junction will fluctuate. 

Thermopneumatic: When the radiation incident on a gas in a chamber increases 
the temperature (and therefore the pressure) of the gas, the chamber expands, moving 
a mirror attached to an external wall. This movement is detected optically. 

Photovoltaic: A change in the number of photons incident on a p-n junction causes 
fluctuations in the voltage generated by the junction. 

Photoconductive: A change in the number of incident photons causes a fluctuation 
in the number of free charge carriers in the semiconductive material. The electrical 
conductivity of the responsive element is inversely proportional to the photon number. 
This change in conductivity is monitored electrically. 

Photoelectromagnetic: Photons absorbed at the surface generate charge carriers 
which diffuse into the bulk and are separated en route by a magnetic field. This 
separation of charge produces an output voltage which fluctuates according to fluc¬ 
tuations in the number of incident photons. 

Photoemissive: Incident photons impart sufficient energy to electrons on the photo- 
emissive surface to liberate them from the material. The freed electrons may be swept 
to an anode, and the resulting current can be monitored to determine fluctuations in 
the number of incident photons. 

11.1.2. Windows. Window materials are primarily used to restrict the spectral 
bandwidth to which the detector is sensitive and to form part of the dewar flask. In 
selecting the window one must consider a variety of material properties. Chapter 8 
describes in detail the various window materials and their properties. 

11.1.3. Apertures. Apertures are often used to limit the field of view of the detector 
in order to reduce photon noise. Section 11.5 of this chapter discusses this subject in 
some detail. 

11.1.4. Dewar Flask. Dewar flasks are used to house the coolant, which reduces 
the detector temperature in order to improve detectivity. The various dewars and 
associated cryogenic systems are described in Chapter 12. 

11.2. Detector Parameters 

The various symbols and units used today for describing the performance of infrared 
detectors are presented in Table 11-1.* Basic noise mechanisms which limit detecting 
ability are listed in Table 11-2 along with a description of the physical process and the 
appropriate mathematical expression. For more detailed information or background 
material on the individual items in these tables, several textbooks [1,2,3] are available 
which treat the subject more thoroughly and also provide further references to the 
archival literature. In addition, two detector state-of-the-art studies [4,5] have been 
published as government publications. 


*Table 11-1 is a compilation of the symbols and units arrived at over the years by workers in the 
infrared community. Especially helpful in preparing the table were Prof. H. Levinstein, Dr. R. C. 
Jones, and W. L. Eisenman. 



460 


DETECTORS 


11.3. Data Enumeration 

This section presents data on commercially available infrared detectors. Of primary 
concern to the system designer are such parameters as: 

1. Detector temperature 

2. Normalized detectivity 

3. Time constant 

4. Resistance or impedance 

5. Spectral response 

6. Noise spectrum 

7. Responsive area 

8. Responsivity 

9. Viewing solid angle 

10. Background temperature 

Data of this kind are available from the manufacturers’ brochures or from either 
of the two cell-testing facilities: Naval Ordnance Laboratory, Corona, California 
(NOLC), and Syracuse (New York) University (SU). NOLC and SU have each pub¬ 
lished a series of reports describing the results of their measurements: Properties of 
Photodetectors [6] and Interim Report on Infrared Detectors [7], respectively. Com¬ 
pilations of data from these series and other sources for each commercially available 
detector are presented in the data sheets at the end of this section.* 

In addition to the parameters listed above, the design engineer must be concerned 
with the effects of environmental factors on detector performance. Data of this kind 
are not available from brochures or from the archival literature; one must rely on 
confirmation from the contractor. The Detector Evaluation and Information Com¬ 
mittee of IRIS (Infrared Information Symposia) has compiled minimum environmental 
specifications (see next page). Most detectors meet these specifications. 


*Three detectors which are not commercially available but are of interest have been omitted. 
They include superconducting bolometers, carbon bolometers, and T1 2 S. At least one other, the 
germanium bolometer, has not reached the state of maturity such that reliable data are available. 



DATA ENUMERATION 


461 


IRIS Environmental Specification #101 [8] 

A. Storage Temperature Requirements 

1. Minimum Range: The range of minimum temperature for detector storage 
shall be +71° C to -55° C. 

2. Accelerated Life Test: The detectors shall meet minimum specification 
requirements within 24 hours after storage for 200 hours at 71° C ± 2°C. 

3. Temperature Cycling Tests (Thermal Shock): Detectors shall meet per¬ 
formance specifications within 24 hours after being subjected seven times (not neces¬ 
sarily consecutive) to the following cycle: The detector temperature shall be raised 
from room temperature to 71° C ± 2° C and held there for two hours; lowered within 
five minutes to —40° C, or below, and held there for two hours; raised within five min¬ 
utes to 71° C ± 2°C and held for two hours; lowered within five minutes to —40°C 
or lower, and held there for two hours; and then raised to room temperature, com¬ 
pleting the cycle. 

B. Humidity Requirements 

The detector shall meet the performance specifications within 24 hours after 
spending ten days in an environment having a relative humidity of (+0%, —5%) 
and a temperature of 38° C ± 2° C. 

C. Vibration Requirements 

The detector element plus package shall meet the performance specifications after 
vibration along all three mutually perpendicular axes from 20 cps to 2000 cps at the 
rate of five minutes per octave with a peak acceleration of 10 g. 

D. Shock 

The detector element plus package shall meet the performance specification after 
being subjected to 18 shocks (three in each of six directions) in accordance with 
Paragraph 4.15.5.1. of Specification MIL-E-5272 [9]. 

E. Microphonics 

Microphonism depends greatly on the method of potting. System requirements 
are very different. Microphonism specifications must be established for the specific 
application requirements of the user. 

F. Acceleration 

Details depend upon system specification requirement. Tests shall be made in 
accordance with Specification MIL-E-5272, Paragraph 4.16.1, when applicable [9]. 

G. Vacuum Environment 

The detector shall meet minimum specification requirements within 24 hours 
after storage for 200 hours at a pressure below 10 ~ 6 torr. This specification does 
not apply to detector elements permanently mounted in a vacuum dewar. 

H. Other 

The following earth environments do not apply to normal applications of infrared 
detectors: salt spray, fungus, rain, sand, dust, immersion test, and explosion-proofing. 


Table 11-1. Detector Parameters 


462 


DETECTORS 


a .& 
1 § 
si 

£ * 
* 05 


CO 

• fH , 
0) 

a> 2 
u cO 

15 

IS 


d> 


<u 

S 5 


<1> A 
X .2 

ci_ a) 

O x 

a +* 
o 


° x 'S 


cO 

a 

cO 

o 


ts 

c 

CO 


3 

^ i 

12 

2 o 
cO cO 
X 


ft ® 
be 

co 5 

£ h c 

in 8 —i -5 

2 ® C! Si CO 

• h ^ d 5 m fc 
N ^ -- 



s js' « ts 



si 

o 

*•>* 

o 

3 

o< 

c*3 

? 

* 

3 

MS. 

cu 

Q 


X 

43 

X) 


X 

XI 

ts, 

c 

a> 


~0 

H 

tJ 


H 

a 

E 


0? 



0) 

J3 

> 

6 

3 

e 

cO 

s 


« * 
Q£ 43 

> 


co 

3 


co 

a> 

(4 


cO 
0) 

« a 

CD 

s 

I II 

ao5 

CO 

II ^ 

eo ^ 

<D w 

U (V 
v 7 s 

I ° 



•o 

1 1 


c«q i -, 

■o 

■S|"Q 


II 

ii 

N 

105 


c 2 

•3 X 

X 43 


i aj 

bo x 
CO £3 

£ .2 
8 cO 


Ills 


XJ 

cO 

s 


CO 


CO 


1?1 
e -3 ® 


O 

c 

43 O. 

•k* 

CD 

E 

&1 


£ 


O) 

Q 

% 

<u 

^ > 
be ;3 


S 6 a * 
£ $ £ 2 
® - cO 

nfl co 

S jj > 

'S' 2 || 

h't 3 (2 
a x *ts 

« O ® 

® CO M « 

<u fe .s £ 

-2 2 -S CO 



0) 

g 

3 

o 

c 

0) 

E 

5 

o 

<u 

be 

co 

43 

'o A 
> C*q 

x ® 
•5 be 

c^ $ 

° 'o 

a > 
° 3 

CO -g 

s ^ 

^ cO 
H 


g -A 

** X 
2 be 

2 | 

y 2 

CO -g 
0) 

be c 
co § 

° ^ 
> 3 

CJ 


o 

XJ 

0) 


o 

XJ 

0) 


■5 -O 

■*“’ 43 

«4H _ 

O S 
.2 n 

43 o 

co £ 


X ^ 
H ^ 


co 

• 

«4 _ C 

^ o ^ 

&} vj 

e -o "3 
§ | 2! 

O CO <ft 
5h w gT 

Oh 


0) 

> 

CO 

a 

a 

CO 

s 


co 

< 


£ 


CD 

O 

C i— 1 

N J 
a s o 
£ 


0) 

o 

c 7? 

•Sias x 

’3 w i£ 
s 

05 


Background The effective temperature of all radi- 

Temperature ation sources viewed by the detector 

(Tb) exclusive of the signal source. 

[°K] 




DATA ENUMERATION 


463 



M 

N 

3 

+ 


3 

§- 

‘S'* 

3 'fs 

I'S 

> co 
, ® 
e * 

5 c 
.§ ? 
3 8 

g 8 

0) "ea 
X o 

~ *c 

CO 3 

43 

^ a 

o M 

© a> 

1 « 

E ^ 

*■« 

| § 

1 * 


3 -£ li 
3 © cd 

ea ^ ^ 

a.2 o 
8 fc*S 

o :> > 

-S co 2 

_ 3 43 

g A 3 

3 a d 
05 

^ OO 
CO 5*2 


■* g 

s s 

E ® 

*3 

. (X) 

3 > 

|1 

“ 8 . 
72 50 
g 3 

X Li 
-L> 

3 3 

.15 

N v_ 
CO O 


c 

0) 

s 

*3 

X 


T3 

'o g 

1b 
t3 | 


3 ai* 


<N 

® 3 

M P T < . 

.5 e 

co cO 

fc § 

II § 

11 o 

a - 

3 
3 

0) 

X 

-L> 

CO 

0 

2 

3 

X 

i* 


3 

e 

•FH 

3 

O 


•c 

a 

CO 

• PH 

Li 

3 

£ 

a 

3 

o 

• rl 

-W 

CO 

• ph 

T3 

CO 

L 

-L> 

3 

TJ 

u 

3 

• pH 

0) 

X 


+ 

-£ 

+ 

Ho 

J 

oo 


a, 

+ 

o 

&h 


+ 

+ 

HO 

J 

co 

O 

u 

+ 

O 


2 

2 

CO 

• pH 

CO 

o 

•pH 

s 

2 

4-> 

8 

IS 


c 

0) 

X 


cO 

8 

oo 


0) 

X 


O 

3 

•pH 

CO 

bo 

o 

73 

0) 

X 


73 

3 

3 


i 

c5 


<4-1 

o 

3 

o 

• f* 

bo 

3 

L 

3 

X 


co 

0) 


CO 

£ < 

TJ V 

g ^ 

® \/ 

<-» V 

V < 

v < 

^ I 

<1 I 

x II 

^ <o 

6 .3 

3 p 
CD O 



3 


O jC 
rri pt? 


CO 


0 ) X 

e * 


3 3 

a <d 

-L> Li 
3 3 

3^ 

3 O 
.2 co 

C — 

■ X 
o 


.81 £ 


fCj 
CD * 

X ^ 


3 

3 

b 

3 

u 

Li 

O 

3 

bo 

3 


JS1 


<4-1 

o 

3 

3 

3 

o 

a 


8 


3 ^ 

O O 

•H 

•3 © 

3 *PH 
Jg 3 

£ I 

^ Jj 

• 2? g 3 

" 3 S3 

E c X 

3 *° p^ 
" 3 § 
3 X 
« X 

H- « 2 

Oh 3 

^ ^ 3 

5 I | 


3 

X 


3 

Li 

3 

bp 3 ,_, 

•£ 3 C ^ 

■e a w e, 

2 E “ 

3 S 
Cu L_| 

o H 


p3 

11) 

&•§ 

o ^ 
3 X 

3* 


3 

.2 

x 

3 

Li 

3 

-L> 

CO 








Table 11-1. Detector Parameters ( Continued) 


DETECTORS 


464 


"a 

§•§ 

■a | 

si 

£ * 
^ fl¬ 



ea 


CO s '"~ 0) 

— „ be 

> cd 

>> ’o 

-u O > 

•I-* ^ 

^ H M 

’« o ® 


£ -5 3 

8-a 


a 

CO 

£ 


2 cd 


0 ) 

> 

a 

cd 

co 

cd 

cu 

£ 

cd 

CO 


c 

o 

** 

O 

3 

O' 

&q 

9 

• «•* 

c 

<e. 

cu 

Q 


o 

u 


8 


CO 


8 

C 

2 

-*> 

8 


1 


§ 


a» 

o 

C 

’3 

be 


tJ 

a) 


A 


E 

V 


II 

1 




* 

ft! 



•o 


5 

!* 



AQ 

«5 


ft! 



I Ch — I a; 

a o « g x> 

o a fl •! +> 

► J .§> TJ<~ 

co cd © 


od 

> 


© co 


S) co C d) cd 
co C 3 c d 


-£ •>- 


CD 
-u 

G 

Cd © 

3 *» 

'S 3 
fc "2 
<2 £ 

*“• d> £; 
~ O Q 

«h C -fd 

III 


C 

0 ) 

T3 


Jt 

cd 


s-“ 

•< _| cd 



fc. 

£ 

cu 

£ 

2 

a 

ft. 


CO 


^ C 

o 10 
*2 *9 

rr? ^ 
CO <d> 


9j 

O 
£ 

CO „ 

s £ 
0 ! 


c 

a> 

a) fc 

bO 5 


cd 


O 


° S 




co 


co 

£ 

>H 

a 

£ 

cd 

o 




(X 

£ 

cd 


cd 

£ 



d> 

> 

a 


„Q O ® 

£ & Q! 


co 

cd 

d> 

£ 

cd 

CO 











DATA ENUMERATION 


465 


<N 


J: S 

73 3 

J 

a i 

£ 6 

« 3 


c 

so 

,2 o 

bp 3 

3 


a 

Ou 

o 

JS 


(V 


« 2 
J3 " 


as 


> 


4^, 

^ CO 

a) e 

" a 


V 

J3 

* 


CD 
3 
00 rs 

0) 3 
£ > 



o 

5 

05 


$ 


g 

05 


9 

CO 

3 

8 

0> 

6 


O) 

> 


0) 

Q 


•« 

k 


s 05 

« •—' 

-»-> i 

3 


Q 


a 8 & 


<u 

6 
• H 
-4-J 

0) 

> 

■ rH 

CO 

3 

a 

CO 

2 

"3 

a 

*C 

'S. 

s 

w 


05 

S 

i 



•9 

«c 

• H 

I 

CO 

Ji 

-t-> 

CO 

3 

6 


T3 

3 

cO 


9 a 

3 r 


* 

Q 


'S 

CC 


* 

Q 


1 — 1 CO 

1 -8 

5 1 

£ 

Q ^ 

T3 


(N 




C 


6 5 g 
S 5 '-s 

8 , 8 2 
“ 5 Jg 



t- 

e» 

a 

+ 


3 

3 _ ^ 
g 3 £ i 

go u - 

1° 







Table 11-1. Detector Parameters ( Continued ) 


466 


DETECTORS 


•*«*» 

•9- 

-3 

o 

‘>4 

co 

c 


•2 

3 

o 

~S 

ft! 


* v 

^ g 

- o 
® • 
as 

c ^ 

O T3 

g-T3 C 
3 C « 

■S 


8 . 

o 


Cfi 

1 

03 

o 




C 

o 

• *•»» 

<3 

3 

Cr 

C*3 

* ««* 
C 
MS. 
o> 

Q 




Oh 

w 

z 



Cu 

W 

z 


Q 


Pu 

W 

z 


II 

00 

as 

Q 


c 

•2 

■>« 

•N» 

c 

MS. 

0> 

Q 




i •* fll 

•3 8 5 
£ S.S 
8 g'S 

‘o * 
G 5P 3 
-= .£ x 

t S’® 

III 


co 
G 

<D 
co 

>-> 

s 

t 

<U <G 
-O 

V 
43 


5 


CJ "O 

o> £ 

O _D 


CO 


T3 

G 

G 


co 

£ 




T3 

•s 

43 

4 a 

C -1 

13 . *g 

4J 3 

3 2 

_ _ O. cn 

G o> 

1 

> 

-S3 

G 

T3 

G 

0 U V 

arc 

aj -C 

1 : 

8 *a fa 

is c 5 

’43 



|5 

ll 

a, ~ 


42 

• *»* 

c 

"O 

2! 

2! 


G rt . 

t 8 


o 

Z 


m 


G 

13 

> 


3 O 
o* Dh 

w ^ 


.< 

b CL 

> w > 

* Z 1 


03 



>> 

-fcj 

>» 

>> •£ cr 

2|qJ 

8 G w * 

Ills 

“1- s 

Cg JT »—> 

S.Q — 
w 

CQ 






DATA ENUMERATION 



vy /-S 

00 Q 
co 

o 

-4-* 
8 

T3 

'S 


O 

c 

I 

T3 

c 

o 

be 

o 

3 

Xi 

u, 

o 


a 

3 

CD 

T3 

e 

a> ■ 

S’h 

C| § 

£ CJ 

05 ^ 

b § 



CO 

3 


0) 

i 

cO 

03 


467 


Q 



II 

C® 

•» 

■< 

* 

Q 



II 



05 

oa 




* 



>» A 

:£ 8 
.5 13 

i c 

% eo 

^ cO 
-2 0) 
CO Sh 
u CO 


o 

X 


I 

co 


3 3 
X X 
Eh * 


2 .SI 

liii 

fH rfi in 


3 3 g 

c 'S t3 

3 <n ►> c 


e 3 

aC® 

3 w 
T3 


G X 


"* CO 


8? 


* 

T3 


N C S 

a -5 _g 

2 3 

£ X 

I 2 


>> 
o 

c 

3 

I 

X 

bo 
- B 

•H 

CO a 

2 o 
■C xs 

-M u 


. T3 
£.£ 

"S 3 

?£ GO 

li S 

a 3 
co x 

co ^ 


co 

Sh 

3 

5 

-o 

Sh 

o 

Pu 


C 

6 


G 
0) 

3 3 
p. X 
o 

3 a 

-C 


c — 

-4^ 


O 00 

5 I 

CO 

Sh 05 

cO _ 

>1 
cO 




'E 

cO 


••H co 
n C 

§ 5 

X 2 

3 Xj 
3 a 

H-> 

O TJ 
3 3 
3 3 3 

o -r: 

3 ' 

ij co 
>> 3 
X5 X5 


'S 


3 J §> I 


JB 




09 C h> 

■ O CJ 


CD "S 

• fh 3 


2 3 


i 

03 


03 h: 


co 

a 

_CJ 

6 

3 


Jj 3 

3 £ 

«Q 

CQ 


as 

* 

Q 


CO 

a 

£ 

CJ 


1 


~ Sh - S* 

S 2 

X 03 ^ a 
3 X.** W 

2 Q 9 e 

o 


Sh 

3 

03^ 
0) ^ 
FH 3 ^ 

3 3 ^ 

£ 5 5 

B 9 Q 

03 


c* 


(fl 




O-i co 


£ 

o 


Peak The wavelength at which detectivity — Depends upon cell 

Wavelength is a maximum. temperature and de- 

p ) tector material used, 

[micron] 






Table 11-1. Detector Parameters ( Continued) 


468 


DETECTORS 


- 

g -c 

•2 1 

tS -2 

I « 

eg * 

^ 05 


'3 

• 

0) 

T3 

*$ 

co 

3 

• H 

I 

3 

o 

* 

s 

o 

T3 

CO 

X 

V 

8 

3 

3 

cd 


3 

a> 

be 

a 

3 

OJ 

G 

3 

-G 

•e 

a> 

4$ 

cd 

a 

3 

S 

• rH 

1 

T3 

CO 

cd 

G 

s 

CO 

aT 

3 

cd 

T3 

3 

S. 

0) 

a 

6 

V 

G 

3 

8 

-a 

3 

a 

d> 

be 

cd 

o 

-g 

3 

3 

-G 

Q 


+-> 

Q 

> 

CO 


c 

o 

*«3 

<3 

3 

Sr 

•pa 

c 

'■e. 

Q 


5 

© 


c*q 

O' 

Q5 


o 

S 

© 


o 

%» 


o 


© 


c*q 

O' 

Q 


e 

o 


3 

'C. 

cu 

Q 


Xi 0) 
2| 
s: 

* 03 


X 

o 

X 

* 


a 

,2 


cd 

X 

I 

<x> 

c 

o 


X 

to 

13 

g3 

<u 

x 3 
H 73 


£ .A 

X a 

5 -S 

3 

3 ° 
8 u 
<2^ 
° 6 
G 

s 

3 

s 

a> 

X 


3 

3 

V 

X 


o a> 


I S 


cd 

G 

<x> 

X 

Eh 


O 

3 * 
a *j 

-g c 
3 a; 
O T3 


V X 

§ s 

3 X 

co 

3 

2 3 

6 « 

o 8 

© 8 
X X 
cd -*» 
G 

a> 

2 jo 

X -g 


a) 

E 

cd 


«c 


«- ^ C 

s-sji 

<b 2 

S *2 “a 
K fc ^ 

CO <A> 


5 

a 

ft. 


£ 


te 

3 

3 

o 


X 

ti r—< 

3 3 

X o G 
© *< O 

«5 w £ 

£ - 


& 

c 


© - 
> 6 

| 1 .2 _ 

I 3 £ 0$ 


&3 

O' 


P ►> 

<u C o 

•||.§S 

i ii? 

^ cyw s 





Table 11-2. Detector Noises 


DATA ENUMERATION 


469 


CO 

C 

o 


K 

> 

K. 

c 

o 


O' 


<J 

£ 


IE 


< 

N 


o 

CL 

N 

'<* 

+ 


3 

3 

■*-> 

o 

3 

02 

s 

3 

CD 

(- 

& 

6 

5 

8 

CO 

3 


0) 

o 

G 

CO 

u 

3 

T3 

3 

8 


II 

O’ 

CO 

3 

N 

§ 

| 

<d 

s 

3 


e 

43 

4-> 


in 

III 


<3 

8 

3 

43 

* 


cO 
O, 
CO 
u 
-»-> 
3 
3 
45 


k o 


£ 

2 

13 

o 

43 

ao 




CO 

co 

a> 

G 

4C 


3 

o 

co 

cO 


b 

•o 


3 

k> 

3 

43 

£ 


§ I 

CO 

c s 

8 8 


o "« 


"o 

81 

£ 

o 

O 

£ 

3 


3 



3 

45 

>» p 

CO 

3 | 


3 <B 
G 73 

-tJ 

4) 

6 g 
° G 

T3 

"8 3 

< 

0.5 


8 

CO 

* M 

C 

-3 

X 


.§ 

co 

£ 

a. 




6 5 Z .5 


.a ^5 



'S G 

3 

s c 

•S 

3 3 

o 

« 43 


O E-> 

3 


4J I* 
^3 O 

1 

G .2 

o 3 
§ | 


43 


2 

3 

-u 

3 

l» 

0) 

a 

e 

3 


3 

G 

O 

• rt 

js 

3 

■8 

S 






470 


DETECTORS 


t- 

<N 


t- 

N 

T)< 

+ 


•o 


IQ5 


c 


S3 


Q 

<1 

v> 

■tt 

O- 

<N 


c 

£ 


O) 


o 

0> 

S3 

Xi 

o 

01 

-fi 

-*-> 

CO 


0) 

Ih 

0) 

X 

£ 


£ 





o 

CO 





DATA ENUMERATION 


471 


GaAs 

Detector Temperature 300°K 



B 

T 

R 

z 

A 

fov 


300°K 
<1 /isec 

~6 x 10^ ft 

in" 2 2 
~10 cm 

180° 

6 volt rms 


10 


watt rms 


max 

Operating mode: photovoltaic 

Limiting noise: 1/f for f <200 cps 
and shot noise for f >200 cps. 

Manufacturer: Philco Corp. 



Chopping Frequency (cps) 


Data Sheet 1 










D* (A, 90) cm(cps) ' watt 


472 


DETECTORS 


Cu-Cu z O 

Detector Temperature 300°K 



T 300° K 

B 


R 

z -ix io 6 n 

A -4 X io' 2 

fov 180° 


max 


.5 x 10 4 vol ‘ rms 
watt rms 


Operating mode: photovoltaic 

Limiting noise: 1/f for f <100 cps 
and shot noise for f >100 cps. 


Manufacturer: Philco Corp. 



Wavelength (p) 



Chopping Frequency (cps) 


Data Sheet 2 







D* (A, 1000) cm(cps) ' watt 


DATA ENUMERATION 


473 


I 


Si* 

Detector Temperature 78°K 



T_ 300°K 

D 

t < 1 to 300 jxsec (often limited by the RC 

value of the sensitive element) 

_ g 

R ~10 fi (for photoconductive detectors) 

7 

Z 50 kft to 10 fi (for photovoltaic detectors) 

A up to 1 cm 2 

fov 180° 

„ , rt 6 volt rms 

K 1U 77- 

A _ watt rms 

in ax 

Operating mode: photoconductive and photovoltaic 

Limiting noise: Johnson noise of the load resistor 
and shot noise for the photovoltaic mode. G-R 
and Johnson noise predominate in the photocon¬ 
ductive mode. 

Manufacturers: Texas Instruments 

Solid State Radiations Inc. 

RCA Ltd. 

International Rectifier Corp. 

* Lenses are supplied by some manufacturers. 



Data Sheet 3 













cm(cps) ' watt' D* (X, 1000) cm(cps) 


474 


DETECTORS 


I 


3 

» 



PbS* 


Detector Temperature 300°K 



T_ 300° K 

B 

r 1000-5000 Msec 

K .5-20 MD/square 


Z 

A 

fov 


max 


10 ® to 10® cm^ 
180° 

-1Q 6 volt rms 
watt rms 


Operating mode: 

Limiting noise: 

Manufacturers: Infrared Industries (IRl) 
Eastman Kodak Co. (EK) 
Electronics Corp. of 
America (ECA) 

Tupper Trent Co. 

Mullanrd Ltd. 

Santa Barbara Research Center 

’Tradeoffs between parameters are available 
upon request from the manufacturer (e.g., t 
may be decreased at the expense of D*). 



Data Sheet 4 






D* (2.5 ii, f) cm (cps) v watt D* (X, 90) cm(cps) ' watt 


DATA ENUMERATION 


475 


PbS* 

Detector Temperature 195°K 
Tg 300°K (b ackground noise insignificant) 

* Tradeoffs between parameters are 
available upon request from the manufacturer 
(e.g., t may be shortened at the expense of D*). 



T 300 U K 

13 

r 500-3000 ^sec 

R .5 to 10 Mfl/square 

Z 

A 10 ® to 10^ cm^ 

fov 180° 

„ _ in 6 volt rms 

X watt rms 

max 

Manufacturers: Infrared Industries Inc. 

Eastman Kodak Co. 

Electronics Corp. of America 
Tupper Trent Co. 

Mullard Ltd. 

Santa Barbara Research Center 








O* (X , f) cra(cps) ' watt D* (X, 90) cm(cps) watt 


476 


DETECTORS 


PbS* 


Detector Temperature 78°K 



T_ 300°K 

D 

t 1000-5000 psec 

R .5-20 MD/ square 

Z 

A 10 ® to 10^ cm^ 

fov 180° 


max 


volt rms 
watt rms 


Operating mode: 


Limiting noise: 

Manufacturers: Infrared Industries Inc. 

Eastman Kodak Co. 

Electronics Corp. of 
America 

Tupper Trent Co. 

Mullard Ltd. 

Santa Barbara Research Center 


"Tradeoffs between parameters are available 
upon request from the manufacturer (e.g., r 
may be decreased at the expense of D*). 



Chopping Frequency (cps) 


Data Sheet 6 








D* (A, 800) cm(cps) ' watt 


DATA ENUMERATION 


477 


PbSe 

Detector Temperature 300°K 



T_ 300°K 

D 

t 1-10 psec 

R ~5 Mfi/ square 

Z 

a , rt -6 2 . ,.0 

A 10 cm to 10 cm 

(ov 180° 

D in 3 volt rms 

A__ watt rme 

max 

Operating mode: photoconductive 
Limiting noise: 1/f noise 

Manufacturers: Eastman Kodak Co. 

Santa Barbara Research Center 


Wavelength (p) 



Chopping Frequency (cps) 


Data Sheet 7 









ia (*<13)013 (oooi 4 x) *a 


478 


DETECTORS 



3 


PbSe 

Detector Temperature 195°K 



T 0 300° K 

T 10-100 4HC 

R >90 MR/square 

Z 

A 10*® cm 2 to 10° cm 2 

fov 


max 




Operating mode: photoconductive 

Limiting noise: 1/f noise 

Manufacturers: Eastman Kodak (EK) 

Santa Barbara 

Research Center (SBRO 


Wavelength (p) 



Data Sheet 8 









DATA ENUMERATION 


479 


PbSe 


Detector Temperature 78°K 



t b SOO°K 

t 10*190 usee 

R .5-50 Mil/square 

Z 

A 10'® cm 2 to 10° cm 2 

fov 90° 


max 


volt run 
watt rms 


Operating mode: photocooductlve 

Limiting noise: l/( noise (or (< 400 epe 
and G-R noise due to lattice vibrations 
(or f >400 cps. 

Manufacturers: Santa Barbara Research Center 
Eastman Kodak Co. 

Infrared Industries Inc. 
Llbrascope Co. 


h 


Chopping Frequency (cps) 


Data Sheet 9 







D* (X, 1000) cm(cps) watt 


480 


DETECTORS 


PbTe 


Detector Temperature 78°K 



B 


T 

R 

Z 

A 

fov 


R. 


max 


300°K 

10-30 /usee 
60-120 MJ1/square 


5 x 10 
180° 

10 5 to 10 


cm^ to 2 x 10 

6 volt rms 
watt rms 


-1 


2 

cm 


Operating mode: photoconductive 

Limiting noise: 1/f noise for f <1000 cps 
and G-R noise due to lattice vibrations 
for f > 1000 cps. 


Manufacturer: 

1 


ITT Laboratories 
Minneapolis-Honeywell 



Data Sheet 10 












D* (* , 0 cm(cp8) watt D* (A, 100) cm(cp8) ' watt 


DATA ENUMERATION 


481 


InAs 

Detector Temperature 300°K 


10 


12 


Theoretical Limit 


V 


10 J 


10 



2 3 

Wavelength (p) 


10 


10 


B 

T 

R 

Z 

A 

fov 


300°K 
<2 peec 

~20 D (requires transformer coupling) 

-1 9 

10 ° to 10 cm 
180° 


~10 


volt rms 

watt rms 


max 

Operating mode: photovoltaic 

Limiting noise: 1/f noise for f <200 cps 
and Johnson noise for f >200 cps. 

Manufacturers: Philco Corp. (provides sap¬ 
phire immersion lens) 
Texas Instruments 
Electro-Opt leal Systems 


10 



10 “ 10 “ 
Chopping Frequency (cps) 


Data Sheet 11 







D* (A, 860) cm(cps) 7 watt 


482 


DETECTORS 


InAs 

Detector Temperature 273°K 

(supplied with thermoelectric cooler and 
an Irtran n immersion lens) 


2 3 

Wavelength (p) 



B 

T 

R 

Z 

A 

fov 


300°K 
<5 /isec 

-150 n 

-2 2 
-10 cm 

180° 

v 3 volt rms 


~10 


watt rms 


max 

Operating mode: photovoltaic 

Limiting noise: 1/f noise for f <30C 
and Johnson noise for f >300 cps. 

Manufacturer: Texas Instruments 



Chopping Frequency (cps) 


cps 


Data Sheet 12 










D* (X, 900) cm(cps) watt 


DATA ENUMERATION 


483 


Te 


Detector Temperature 78°K 




T 

R 

Z 

A 

fov 


max 


300°K 
60 p sec 

1000 ohms/square 

>2 x 10' 3 cm 2 
180° 

^^4 volt rms 
watt rms 


Operating mode: photoconductive 

Limiting noise: 1/f noise for f <1000 cps 
and G-R noise due to lattice vibrations 
(phonons) for f >1000 cps. 

Manufacturer: Minneapolis-Honeywell 


Wavelength (p) 



Data Sheet 13 










(sdD)U 13 (006 *Y) *a 


484 


DETECTORS 


InSb 

Detector Temperature 300°K 



T_ 300°K 

O 

r <.2 nsec 

R ~15 ohms/square 

Z 

2 2 

A (0.05 x 0.05) cm to (0.2 x 0.5) cm 

fov 180° 

_ _ volt rms 

~ 5 —n- 

\ watt rms 

max 

Operating mode: photoelectromagnetic or 
photoconductive 

Limiting noise: Johnson noise 

Manufacturers: Block Associates 

Minneapolis- Honeywell 
Radiation Electronics Corp. 
Texas Instruments 


Wavelength (m) 



Data Sheet 14 







DATA ENUMERATION 


485 


InSb 

Detector Temperature 195°K 



Chopping Frequency (cps) 


Data Sheet 15 






D* (X , l) cm(cps) ' watt D* (X, 1000) cm(cps) watt 


486 


DETECTORS 


I 


InSb 

Detector Temperature 78°K 



T_ 300°K 

B 

r -1 psec 

R 1-20 KD/square 

Z 

-3 2 

A* as small as 10' cm 

fov 180° 


R X 


max 


1Q 4 volt rms 
watt rms 


Operating mode: photoconductlve 

Limiting noise: G-R noise due to background 
photons to lattice vibrations 


Manufacturers: Texas Instruments 

Minneapolis-Honeywell 

Santa Barbara Research Center 

Raytheon 

‘Areas as small as 10~^ have been constructed 
by the Chicago Midway Laboratories (now the 
Laboratory of Applied Science) 


Wavelength (p) 



Data Sheet 16 






D* (A f) cm(cps) 7 watt" 1 D* (A, 900) cm(cps) 


DATA ENUMERATION 


487 



InSb 


Detector Temperature 78°K 




T 

K 

z 

A 

fov 


"x 


max 


300° K 
<1 Msec 


200 to 10 MB 

,--3 2 . 2 

10 cm to .3 cm 

180° 

10 4 to 10 5 

watt rms 


Operating mode: photovoltaic 

Limiting noise: 1/f noise for f <300 cps 
and shot noise for f > 300 cps arising 
from the generation of minority carriers 
due to photon and phonon excitation. 

Manufacturers: Texas Instruments 

Philco Corp. 

Networks Electronics Corp. 

Radiation Electronics Corp. 


Wavelength (p) 



Data Sheet 17 






488 


DETECTORS 


GerAuSb* (n-type) 

Detector Temperature 78°K 



300°K 

20-800 Msec 
2-20 Mfi 

10‘ 3 - 10' 1 cm 2 
- 110 ° 

, rt 6 volt rms 
■ 

watt rms 

max 

Operating mode: photoconductive 

Limiting noise: 1/f noise for f <200 cps and 
G-R noise due to lattice excitations (phonons) 
for f >200 cps. 

Manufacturer: Philco Corp. 

‘Manufacturer will provide a lens for optical 
gain. 





Chopping Frequency (cps) 


Data Sheet 18 






D* (X, 1000) cm(cps) ' watt 


DATA ENUMERATION 


489 


Ge.AuSb (p-type) 


Detector Temperature 78°K or 60°K 



T_ 300°K 

D 


r <1 psec 

R .1-5.0 Mfi/square at 78°K 

Z 

A (.05 x .05) cm^ to (0.5 x 0.5) cm^ 

fov 60° 


max 


1( j4 volt rms 
watt rms 


at 78°K 


Operating mode: photoconductive 

Limiting noise: 1/f noise for f <100 cps and 
G-R noise due to lattice excitations (phonons) 
for f >100 cps. For detector temperatures 
<60°Kthe G-R noise has a significant photon 
noise component. 

Manufacturers: Philco Corp. 

Westlnghouse Research Laboratories 
Santa Barbara Research Laboratory 
Raytheon 


Wavelength (p) 



Data Sheet 19 








DETECTORS 


Ge:Cu 

Detector Temperature ~4°K 



T 0 300°K 


T 

R 


Z 

A 

fov 


max 


<1 nsec 

.02 to 20 Mft square (depends on 
bias voltage and background tem¬ 
perature) 


2 2 
(.05 x .05) cm to (0.5 x 0.5) cm 

26°, 60°, 90° 

10 4 ,0 10 6 ! 2 “£”£ 

watt rms 


Operating mode: photoconductive 

Limiting noise: 1/f noise for f <100 cps 
and G-R noise due to background photons 
and due to lattice vibrations (phonons) 
for f > 100 cps. 

Manufacturers: Texas Instruments 

Santa Barbara Research Center 
RCA 

Eastman Kodak Co. 


Mil 1 




i 

i i l i i 111 i 

1 1 1 1 1 II1 1 1 

1 1 1 1 11 

10 

10 2 

10 3 

10 


Chopping Frequency (cps) 


Data Sheet 20 










Relative S/N D* (X, 900) cm(cps) watt 


DATA ENUMERATION 


491 


Ge:Cd 


Detector Temperature 20°K 



0 10 20 30 

Wavelength (cps) 



T_ 300°K 

O 

t <1 psec 

R 

Z 

2 2 

A (.05 x .05) cm to (0.5 x 0.5) cm 

fov 60° (other sizes are available) 

max 

Operating mode: photoconductive 

Limiting noise: presumably 1/f noise 
for smaller chopping frequencies and 
G-R noise due to background photons 
for larger chopping frequencies. 

Manufacturer: Raytheon 


Data Sheet 21 









D* (X, 900) cm(cps) watt 


492 


DETECTORS 


Ge:Hg 

Detector Temperature ~4°K 




T 300°K 

D 

t <1 /xsec 

R a function of the amount of background 

radiation when the detector is photon 
noise limited. For the conditions spec¬ 
ified here, R is in the range 40-400 kft. 

Z 

A (.05 X .05) cm^ to (0.5 x 0.5) cm^ 

fov *150° (can range from ~8° to 180°) 

„ , rt 6 volt rms 

K. -77- 

X watt rms 

max 

Operating mode: photoconductive 

Limiting noise: 1/f for f £100 cps and 
G-R noise due to background photons 
for f > 100 cps 

Manufacturers: Texas Instruments 

Santa Barbara Research Center 
Raytheon 



Detector Temperature (°K) 


Data Sheet 22 












HUM (sdo)uio (006 ‘Y) *Q 


DATA ENUMERATION 


493 


Ge:Zn 

Detector Temperature 4°K 



0 10 20 30 40 


T 300°K 

B 

r < .01 jisec 

R .2 to 20 MJ1/square 

Z 

A (.05 x .05) cm^ to (.5 x .5) cm^ 

fov 60° 

R X 

max 

Operating mode: photoconductive 

Limiting noise: presumably 1/f noise limited 
for f < 100 cps and G-R noise due to back¬ 
ground photons for f >100 cps. 

Manufacturer: Perkin-Elmer Corp. 


Wavelength (/i) 


Data Sheet 23 



D* (X, -) cm(cps) watt 


494 


DETECTORS 


HgTe [5% ZnTe, 5% CdTe] 

Detector Temperature 500°K 



B 

T 

R 

Z 

A 

fov 


300°K 

*2 /isec 
*5 ft 

minimum (.05 x .05) cm 
180° 

volt rms 


R(500°K) *.05 

watt rms 

Operating mode: photoelectromagnetic 
Limiting noise: Johnson 
Manufacturer: Minneapolis-Honeywell 


Data Sheet 24 




DATA ENUMERATION 


495 


Ge-Si:Zn 


Detector Temperature 50°K 



T_ 300°K 

D 

r <1 fisec 

R ~3 X10 6 ft 

Z 

A 

fov 20° 

^Amax 

Operating mode: photoconductlve 

Limiting noise: for chopping frequencies 
higher than 100 cps G-R noise due to 
photons and lattice excitations (phonons) 
predominate. 1/f noise becomes signifi¬ 
cant for f < 100 cps. 

Manufacturer: RCA 


Wavelength (/i) 



Data Sheet 25 





D* (X, 900) cm(cps) / watt 


496 


DETECTORS 


Ge-Si:Au' 


Detector Temperature 50°K 



T_ 300°K 

B 

t <1 /isec 

R ~40 x 10 6 n 

z 

A 

fov 70° 

R X 

max 

Operating mode: photoconductive 

Limiting noise: presumably 1/f noise for 
f <100 cps, G-R noise due to photons 
and lattice excitations for f >100 cps. 

Manufacturer: RCA 


Wavelength (jx) 


Data Sheet 26 



D* (X, —) cm(cps) watt 


DATA ENUMERATION 


497 


InSb Bolometers* 

Detector Temperature (see table below) 



T 0 300° K 

t <psec 

R 

Z 

A 

fov 

«x 

max 

Operating mode: photoconductlve 

The spectral response may be tuned from 
60 u-300 jx by varying the Intensity of 
the magnetic field 

Limiting noise: unknown 

Manufacturer: Mu Hard Ltd. 


Detector 

Operating 

Temperature 

(°K) 

Magnetic 

Field 

Gauss 

A 

4.2°K 

14 K 

B 

4.2°K 

— 

C 

1.8°K 

5.5 K 


Data Sheet 27 








498 


DETECTORS 


Thermistors* 

Detector Temperature 300°K 



t- 

/<" 



B 

T 

R 

Z 

A 

fov 


max 


300°K 

800 to 8000 *xsec 
10 5 to 10 6 


10~ 4 to 3 x 10 * cm^ 
180° 

10 2 to 5 x 10 3 YO}Lnns 

watt rms 


Operating mode: bolometric 

Limiting noise: Johnson noise for frequencies 
larger than about 40 cps and 1/f noise for 
f <40 cps. 

Manufacturers: Barnes Engineering 

Servo Corporation of America 
Polan Industries 

* Note 1: immersion lenses are available 
which provide a factor of 3 increase in D*. 
Note 2: in general, tradeoffs can be obtained 
between D* and time constant (e.g., D* = 

8 x 10® rty2, where t is in seconds). 



Data Sheet 28 





DATA ENUMERATION 


499 


Thermocouples 

Detector Temperature 300°K 



T 300°K 

D 

t ~10-20 msec 

R 5-15 ft 

Z 

A (.01 x .1) cm^ to (.03 x .3) cm 

fov 180° 

R volt rms 

X watt rms 

max 

Operating mode: thermovoltaic 

Limiting noise: Johnson noise 

Manufacturers: Perkin Elmer Corp. 

Eppley Laboratory 
Charles Reeder Co. 
Beckman Instruments 
Farrand Corporation 



Chopping Frequency (cps) 


Data Sheet 29 









D* (X , f) cmfcps) 17 watt' 1 D* (X, 20) cm(cps) watt 


500 


DETECTORS 



Golay Cell* 

Detector Temperature 300°K 

T. 


B 


T 

R 

Z 

A 


} 


300°K 
~10 msec 


this device uses a pneumatic circuit 
coupled to a photoemissive detector. 


fov 

*X 


~10 


max 


4 volt rms 
watt rms 


Operating mode: thermopneumatic 
Limiting noise: 

Manufacturers: Eppley Laboratory, Inc. 

* Note : window materials such as diamond 
and quartz are available. 



Data Sheet 30 








TEST PROCEDURES 


501 


11.4. Test Procedures 

Experimental procedures to provide the necessary information for proper detector us¬ 
age are presented. Most of the experimental detail supplied here is similar to that 
established at the Naval Ordnance Laboratory, Corona, California [10], and Syracuse 
University [11]. These facilities have been sponsored by the Armed Services to provide 
up-to-date quantitative measurements on all types of photodetectors in the spirit of 
a standards laboratory with respect to the experimental procedures undertaken and the 
data provided. Attention should also be called to the standardization report by Jones 
et al. [12], which crystalizes the thinking of British, American, and Canadian scientists 
on the subject of testing and of describing test results. 

11.4.1. Determination of NEP. The circuitry used for measurement of V s , rm» and 
Vrms and for the determination of optimum bias is shown in block form in Fig. 11-1. 
The important components of this circuitry are the infrared source (blackbody), the 
preamplifier, and a wideband wave analyzer. The blackbody emitter has precision 
temperature controls. A standard technique used in test procedures is to make meas¬ 
urements with a blackbody set at a temperature of 500° K. The source is mechanically 
modulated by a disc-type chopper. Generally, this chopper is arranged with two speeds 
to provide radiation modulated at 90 cps or 900 cps. This chopped radiation generates 
an electrical signal in the detector which is amplified and then measured with the 
harmonic wave analyzer. The wave analyzer is also used to determine the noise level 
by obtaining a reading when the detector is shielded from the chopped radiation. 



Fig. 11-1. Block diagram of test circuit used for making measurements. 


The signal voltage and noise voltage for a photoconductive detector are determined 
as a function of bias voltage. For most detectors, chopping frequency is not a significant 
factor in determining optimum bias. Figure 11-2 shows a typical bias graph with plots 
of signal and noise voltage versus bias current. This graph is typical of those supplied 
by the Naval Ordnance Laboratory. The optimum bias point is determined from such 
a graph and used in all subsequent measurements for that particular detector. Typical 
circuitry of the cell bias and match are shown in Fig. 11-1, and drawn schematically 
in Fig. 11-3. Two sets of input leads are shown from the detector to the match. One 
set is for the bias current used with thermistors, photodetectors, and the like; the 
other set is for photovoltaic and PEM detectors as well as thermocouples. Photovoltaic, 
PEM, and thermocouple detectors usually require transformer coupling to the pre¬ 
amplifier because of their low impedance. It is important to use a transformer whose 
impedance can be varied to insure that the equivalent noise input resistance of the 
preamplifier is transformed to an impedance lower than the impedance of the detector 
being tested. 




































502 


DETECTORS 




BIAS CURRENT (/lamp) 


Fig. 11-2. Determination of optimum bias. 


Photoconductive 
Test Leads 



Photovoltaic 
Test Leads 

Fig. 11-3. Cell bias and impedance matching circuit 
for infrared detectors. 


The simplest kind of circuitry associated with the photoconductive detector is shown 
in Fig. ll-4(a) and consists simply of a bias battery supply in series with the photo¬ 
conductive detector and a load resistor, Rl. The signal is taken off the load resistor 
and fed through a capacitor to a preamplifier. The voltage across the load resistor 
is given by 


V L = V- 


Rl 


Rl ■+■ R 


( 11 - 1 ) 











































TEST PROCEDURES 


503 



(a) (b) 

Fig. 11-4. Bias circuits for 
photoconductive detectors. 


The change in voltage across the load resistor produced by the modulated radiation is 
found by differentiating this equation with respect to the resistance of the cell. It 
then follows that 


y L = V —R l ^_ 

R l + R R 


( 11 - 2 ) 


When the load resistance is much larger than the detector resistance, a constant bias 
current condition exists. Maximum signal voltage is obtained when the load resistance 
is the same size as the detector resistance; extended frequency response is obtained 
for small values of load resistance. This latter requirement usually appears with 
high-resistance fast detectors. In this situation capacitive effects become important, 
and in order to match the response-time capability of the detector it is necessary to 
use a small load resistor, which produces a reduced signal amplitude but flat frequency 
response over a wider frequency range. 

A modification of this simple circuit is shown in Fig. 11-4(6). It involves placing 
a dc load resistor in series with the detector and an ac load resistor (r) across the detector 
through a coupling capacitor. The output is fed to the preamplifier from the ac load 
resistor. The effect of this type of circuitry is to allow changes in the load resistor 
without influencing the photoconductor bias current. These changes are necessary 
to determine the optimum bias current. 

In making any measurements, it is important that the preamplifier noise be less 
than the detector noise. Two types of noise must be considered with respect to the 
preamplifier: (1) an effective series noise; (2) an effective shunt noise. 

The series noise is experimentally determined by shorting the input to the preampli¬ 
fier and then noting the output noise voltage. The shunt noise is determined by open¬ 
ing the input circuit and recording the output noise level. Then, starting with large 
values of resistance, a sequence of resistors of decreasing value is placed across the 
input to the preamplifier and the output noise level noted. The value of resistance is 
reduced to a point where a change in the noise output from the open-circuit condition 
of the preamplifier is recorded. One must then insure that the detector resistance is 
a smaller value than this to provide a detector noise greater than the preamplifier 
noise. In this condition the detector resistance is greater than the series resistance 
of the preamplifier, but less than the shunt resistance. This is referred to as a detector- 
noise-limited condition. This requirement becomes difficult to meet when one is forced 
to deal with very-high-impedance detectors (higher than 15 or 20 megohms). Other¬ 
wise, the problem of shunt noise is not serious, and one usually finds that the series 
noise requires the most caution. 

The primary purpose in measuring signal and noise voltages with the equipment 
of Fig. 11-1 is to determine the photodetector’s noise equivalent power (NEP). This 
















504 


DETECTORS 


can now be done since the power density of the radiation from the blackbody that falls 
on the detector is known. This value can be calculated, starting with the Stefan- 
Boltzmann law. The power density H from a source of radiance AT at a distance X 
to the detector is the detector irradiance: 


X 2 7T X 2 


(11-3) 


where A s is the source area, N is the radiant flux emitted by the source per unit area 
per unit solid angle, and is equal to Win (W = radiant emittance). For a circular 
source aperture of diameter D s , the power density is 


W 7 tDs 2 W As 
7 r 4X 2 7 t X 2 


(11-4) 


and therefore NEP is given by 


NEP = 


HA d 


s t rms 


/Vn, 


rms 


W /D s\ ^ V n, rms 

4 lx/ F i>rms 


(11-5) 


The gain of the circuitry used to determine NEP is checked with an oscillator and a 
microvoltmeter connected to the input of the preamplifier. The noise bandwidth of the 
system is determined by measuring the Johnson noise generated in a wire-wound 
resistance as 

kf=~e 2 /4kTr (11-6) 

where k is the Boltzmann constant, T is the absolute temperature, r is the resistance, 
and e 2 is the mean square voltage fluctuation. The signal-to-noise ratio of a detector 
at a given bias current is generally independent of the load resistance. However, 
as shown by Eq. (11-2), the signal voltage, and correspondingly the noise voltage, 
are functions of the load resistor. Since different applications may require different 
load resistors, a listing of detector signal and noise measurements must include the 
value of the load resistance used in making the measurements. 

11.4.2. Time Constant. Speed of response information is usually provided in one 
of two forms. They are: (1) a plot of response versus frequency from which a detector 
time constant can be estimated, and (2) the photodecay characteristic after removal 
of a photoexcitation source. Information of type (1) is generally obtained by amplitude 
modulation of radiation from- an infrared source irradiating the detector, and varying 
the frequency of modulation; type (2) is obtained by observing the signal wave shape 
of the photodetector response to periodic pulses of light. Systems for making meas¬ 
urements to provide the two types of information are described below. 

11.4.3. Frequency Response. Frequency response is usually measured with a 
metallic-disc light chopper. The disc is ringed with slits spaced symmetrically, so 
that the opaque region and the slit region have the same width. 

The modulation frequency is given by the spinning rate of the disc multiplied by the 
number of slits in the disc. The higher the frequency of modulation required, the 
higher the spinning rate, and/or the greater the number of slits cut in the disc. In¬ 
creasing the number of slits results in slits of decreasing width (for any one size disc), 
until eventually an optical system is required to image down the infrared source onto 
the slit. The radiation passing through the slits is then focused onto the detector. 
For low-frequency operation, sinusoidal modulation can be obtained by proper selection 
of the chopper opening [13]. 

The frequency response was determined at Corona [10] by a variable-speed chopper, 
giving a frequency range of 100 to 40,000 cps. Radiation from a Nemst glower is 





TEST PROCEDURES 


505 


sinusoidally modulated by the chopper, and is usually filtered by a selenium-coated 
germanium window. The signal from the detector is measured by putting the output 
of a cathode follower and a preamplifier into the y-axis input of an oscilloscope. An 
incandescent tungsten source is simultaneously modulated by the chopper, and activates 
a photomultiplier whose signal is fed into a preamplifier and a tachometer; the lat¬ 
ter s output is proportional to frequency and is put on the x axis of the oscilloscope. 
The oscilloscope display is photographed as the chopper slows down from its maximum 
speed. Syracuse University, using a wheel cut with 1400 circular holes spinning at 
a rate of 10,000 rpm, obtains a maximum chopping frequency of 240,000 cps. This 
equipment uses a Globar as the light source and an As 2 S 3 (arsenic trisulfide) lens 
to focus the source onto the slit. For photodetectors whose response can be described 
by 


V. 

(1 + g> 2 t 2 ) 1/2 


_ V s , rms |/= 0 

* s, rms 


(11-7) 


this high-frequency chopping rate permits an evaluation of time constants as short 
as 0.5 Msec. Typical frequency-response data reported from NOL, Corona, is shown 
in Fig. 11-5. 



Frequency response measurements may also be obtained by using an injection laser 
as the light source. The injection current may be modulated sinusoidally at frequencies 
up to the gigamegacycle region — sufficient for determining time constants shorter than 
1 fisec. However, the laser must be calibrated in terms of output intensity vs. fre¬ 
quency to insure that the frequency dependent of the electrical output of the detector 
is due to the detector and not to the source. Several manufacturers are now using this 
technique to determine time constant. 

11.4.4. Pulse Response. Another approach to the measurement of speed of response 
is a direct measurement of the decay or rise characteristics of the detector. For detec¬ 
tors with slow response and high sensitivity, it is fairly easy to design a mechanical 
light chopper with sufficient speed so that the dynamic characteristics measured belong 
to the photodetector, and not to the chopper. However, when one is dealing with photo¬ 
detectors whose response times are less than 1 nsec, normal procedures in making 
this measurement become difficult. To measure the decay or rise characteristics of 
the detector requires a light source whose rise or fall time is approximately 1/10 the 
time that is to be measured. Optical spinning mirror systems can provide such rapid 
rise-and-fall light-pulse time. A rather simple arrangement is shown in Fig. 11-6. 





506 


DETECTORS 



Fig. 11-6. Simple spinning mirror for periodic 
light pulse generation. 


A collimated beam of light is deflected by a rotating mirror. A decollimating mirror 
which focuses the infrared radiation on the detector is placed a distance X away from 
the mirror. The rise time of the light pulse is the time it takes the leading edge of 
the pulse to fill the decollimating mirror, and the fall time is the time required for 
the trailing edge of the light beam to move off of that same mirror. The velocity with 
which the light ray moves across this mirror is given by the distance between the 
spinning mirror and the decollimator, multiplied by the angular velocity of the spin¬ 
ning mirror. The rise time and decay time, assuming a symmetrical light pulse, are 
equal to each other, and to the width of the decollimator divided by the velocity. Obvi¬ 
ously, by making X sufficiently large, the rise and fall times can be made shorter, but 
generally at the expense of decreasing intensity at the detector. The energy may 
be increased by the use of a cylindrical mirror which compresses without affecting its 
width. Light pulses with rise and decay times of about 50 nanoseconds have been 
generated with this technique, using a mirror spinning at 10,000 rmp. 

Another useful spinning-mirror technique is that described by Garbuny et al. [14]. 
The method consists of surrounding a rotating multisided mirror by a set of stationary 
mirrors (see Fig. 11-7). This assembly is so adjusted that the collimated light from 
the source is repeatedly reflected between the central and the stationary mirrors. 
Each face of the mirror rotating with angular velocity a> adds 2N to the rotational 
speed of the emerging light beam. If D is the width of a slit in the image plane, and 
is less than the width of the light beam 8, the rise time and the fall time of the pulse 
are given by 


and 


D 

~ 2Na>X 


Tdecay 


8 — aD 
2 NojX 


( 11 - 8 ) 

(11-9) 


where N is the number of faces on the rotating mirror and X its distance from the image. 
By using a multisided spinning mirror to obtain high tangential velocities, it is pos¬ 
sible to substantially reduce the radial distance from the spinning mirror to the detec¬ 
tor over that required in a construction like Fig. 11-6. Using mirror optics for col¬ 
limating the light source permits any infrared emitter to be used. With a 0.5-mm 
wide sensitive element, a spinning-mirror rotation rate of 10,000 rmp, X = 1 m, and 
N = 6, rise times of 30 nanosec are readily available. Using a turbine-drive motor 
system to spin the mirror, rotating speeds as high as 3000 rps can be obtained, so that 
pulse rise and decay time of less than 1 nanosec are possible. 

Square pulses of radiation may also be obtained by modulating injection lasers 
with square pulses of current. In the fashion described above, the rise time and 
decay time of the detector can then be observed. 





TEST PROCEDURES 


507 



I 

i 

i 

i 

i 


Infrared Emitting 
$ Source & Slit 

-f— 

I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 

Collimating Mirror 



Infrared Detector 


Fig. 11-7. Spinning mirror system for periodic 
light pulse generation in the millimicrosecond 
range. 


Conditions exist where signal-to-noise ratios of close to or less than unity must be 
measured. Examples of such cases follow. (1) Examination of fractional-microsecond 
signals from high-impedance detectors. The measurement technique here requires 
the ac loading of the detector (see Fig. 11-46) with a resistance low enough to provide 
flat frequency response over the spectrum of interest. (2) Examination of fractional- 
microsecond signals from low-impedance detectors such as indium antimonide. This 
measurement is difficult because the noise level of a wideband preamplifier is higher 
than that of the detector. (3) Reproduction of low-level signals caused by low-level 
radiation sources, as for wavelength-dependent measurements. 

A device has been developed which makes measurement in these cases readily pos¬ 
sible, and with signal-to-noise ratios less than 1. This device (often called a synchro¬ 
nous detector) applies a sampling technique and integration (or averaging) to the 
direct measurement of the shape of periodic noise-limited waveforms. This may be 
compared to the usual coherent detector which can be described with reference to 
Fig. 11-8. The signal wave shape is periodic, and is triggered in the same manner that 
would be required for good high-speed oscillographic reproduction; the noise is random. 
The interval At represents the "on” time of an electronic switch, during which the 
signal and noise voltage is fed directly into an integrator. By sampling successive 
intervals and averaging, it is possible to reduce the noise-voltage fluctuation observed 
at the integrator output without affecting the signal level. Quantitatively, the noise- 
voltage fluctuations are reduced by 1/VN, where N is the number of observations made 
during the measurement. The signal-to-noise voltage ratio is then improved by the 
square root of N. If At is made small compared to the signal transient time, and is 
slowly and uniformly retarded in time with respect to signal onset, an accurate chart 
record of the signal wave shape may be produced [15]. 


508 


DETECTORS 



Fig. 11-8. Signal, noise, and gate 
relationship in waveshape recorder. 


11.4.5. Spectral Response. Measurements of the wavelength dependence of infra¬ 
red photodetectors are generally made with an experimental setup like the one illus¬ 
trated in Fig. 11-9. 

The energy flux from the exit slit is measured at each wavelength with a thermopile 
or thermocouple. As the wavelength is changed, the energy falling on the thermo¬ 
couple is rasied or lowered to a convenient value by opening or closing the entrance 
slit of the monochromator, with the middle and exit slits usually remaining fixed. 
Once this level is set, the energy flux is allowed to fall onto the detector, and the response 
is then obtained. A typical relative response curve from Corona is shown in Fig. 11-10. 
Generally, the chopping rate of the light input to the monochromator is 10 to 13 cps, 
compatible with the response characteristics of the thermocouple. However, since 
most photodetectors show considerable improvement of NEP at higher chopping rates, 
it is advantageous when possible to modulate the spectral radiation at frequencies 
of about 200 cps. At Syracuse, the chopper is operated at 208 cps, and the detector 
signal is measured by feeding it through a filter of 30-cps bandwidth tuned to 208 cps 
to a preamplifier, and then to a vacuum-tube voltmeter. At the low chopping frequency, 
the detector signal is fed directly into the amplifying system of the monochromator. 



Fig. 11-9. Block diagram test circuit used 
for measuring detector response. 










































TEST PROCEDURES 


509 



Along with the measured relative spectral response curve, it is important that the 
detector user be provided with an absolute calibration sufficiently universal that the 
spectral dependence of figures of merit such as NEP and D* can be readily derived. 
The information available from the measurements of NEP and relative spectral re¬ 
sponse, and the theoretical law for blackbody spectral radiation distribution, are 
sufficient to provide the absolute calibration. 

Absolute spectral measurements may be obtained using a calibrated thermocouple. 
However, the detector being measured must be placed at a point in the monochromatic 
beam, where it receives the same energy as the calibrated thermocouple. Since this 
is difficult to do in practice, a more suitable method is common. In this method use 
is made of the relationship 

D*(kfo) = KR x ' (11-10) 

where Rk is the relative response and if is a proportionality constant expressed as 


D*(TBB,fo ) 


2 F *i R *i 


( 11 - 11 ) 


Here, Fx, is the fraction of energy in a particular wavelength interval (AX) of the 
spectrum of the blackbody used for detectivity measurements and corrected for the radi¬ 
ation emitted by the chopper which is at 300° K. Values of F\ t for wavelengths between 
1 and 30/a are given in Table 11-3. More general tables are given by Lowan and 
Blanch [16]. The constant K must be evaluated for each detector. 





510 


DETECTORS 


Table 11-3. Energy Fractions for a 500° K 
Blackbody in a 300° K Surrounding Medium 


Wavelength Interval (/x) 

1-1.5 

1.5- 2.0 
2.0-2.5 

2.5- 3.0 
3.0-3.5 

3.5- 4.0 
4.0-4.5 

4.5- 5.0 
5.0-5.5 

5.5- 6.0 

6 . 0 - 6.5 

6.5- 7.0 
7.0-7.5 

7.5- 8.0 
8.0-8.5 

8.5- 9.0 
9.0-9.5 

9.5- 10.0 
10.0-10.5 

10.5-11.0 

11-12 

12- 13 

13- 14 

14- 15 

15- 16 

16- 17 

17- 18 

18- 19 

19- 20 

20 - 22 

22-24 

24-26 

26-28 

28-30 


Energy Fraction 
(500° K Blackbody) 

7 x lO 6 
3.7 x lO" 4 
0.0032 
0.012 
0.024 

0.038 

0.050 

0.058 

0.062 

0.063 

0.061 

0.058 

0.054 

0.050 

0.045 

0.041 

0.037 

0.033 

0.029 

0.027 

0.045 

0.035 

0.029 

0.022 

0.019 

0.015 

0.013 

0.011 

0.0084 

0.015 

0.0097 

0.0072 

0.0058 

0.0029 


11.4.6. Noise Spectrum. The noise-voltage spectrum is obtained with the system 
described in Section 11.3.1. However, the light source is removed and the noise 
voltage is obtained by simply reading the voltage at the wave analyzer. A typical 
plot of noise spectrum is shown in Fig. 11-11. 

11.4.7. Sensitivity Contours. If a microscopic ray of light is projected onto the 
surface of a photodetector, and the photoresponse recorded as a function of the ray’s 
position, it is found that the photoresponse generally changes with the ray’s position. 
The surface of the detector is thus rarely uniform in its photoresponse. The film 
detectors (lead compound family) are the worse offenders in this regard. If a graph 
of photoresponse versus light-ray position is made, and points of equal photoresponse 
are linked together, the resultant plot provides a "sensitivity contour,” illustrated by 
Fig. 11-12. 


TEST PROCEDURES 


511 


> 

w 

co 

O 

z 



FREQUENCY (cps) 


Fig. 11-11. Detector noise spectrum. 



Direction 

-4 - 

of Scan 


Fig. 11-12. Sensitivity contour for typical PbSe cell. 


At Corona, the experimental arrangement to obtain sensitivity contours uses a 
microtable which allows the cell to be moved a small measured amount. The table 
is linked through a system of gears to a plotting table which gives an increase in the 
scale up to 36:1. The exciting radiation is from an incandescent tungsten bulb chopped 
at 90 cps; it is passed in reverse through a microscope such that a spot 0.066 mm in 
diameter is focused onto the detector. As the detector is moved beneath this radiation, 
the relative response at 10% intervals is noted on the plotting table. Lines connecting 


















































































512 


DETECTORS 


equal points of sensitivity are then drawn to obtain a plot such as Fig. 11-12. This 
light-probe technique is also important for its utility in fundamental research programs 
on detector materials, where it is used in studies of diffusion length, time constant, 
and mobility [17]. 

11.4.8. General Comments. A summation of data necessary to evaluate a detector 
is shown in Fig. 11-13, which consists of a typical data sheet from an NOLC report. 
The definitions of the various parameters are listed in Table 11-1. Data on specific 
detectors is in Section 11.2. 


TEST RESULTS 

CONDITIONS OF MEASUREMENT 

R (volts, watt) 

(500, 90) 

4.1 x 10 4 

Blackbody temperature 
(°K) 

500 

H (watts/cps 1 / 2 • cm^) 
(500, 90) 

Q 

Blackbody flux density 

9.0 

8.8 x 10 

(pwatts/cm^, rms) 
Chopping frequency 

90 

P (watts/cps 4 / 2 ) 

(500, 90) 

5.6 x 10" 11 

(cps) 


Noise bandwidth (cps) 

5 

D* (cm-cps 1 / 2 ) 

1.4 x 10 9 

Cell temperature (°K) 

197 

(500, 90) 


Cell current for 

7.0 

Responsive time 

26 

90-cps data (pa) 

constant (psec) 

r a 

Cell current for 

D * 

mm 

20.0 

max 

9.1 



R bb 


Load resistance (ohms) 

2.5 x 10° 

Peak wavelength (ju) 


Transformer 

_ 

2.2 

Relative humidity (%) 


Peak detective modu- 

4 x 10 3 

16 

lation frequency (cps) 

Responsive plane 


D* (cm cps 1//2 /watt) 
mm ' ' 

1 0 

(from window) 


2.8 x 10 

Ambient temperature 
(°C) 

24 

CELL DESCRIPTION 

Ambient radiation 
on detector 

297°K only 

Type 

PbSe (evap.) 



Shape of sensitive 
area (cm) 

0.038 x 0.168 



Area (cm 2 ) 

6.3 x 10" 3 



Dark resistance 

0 



(ohms) 

1.64 x 10 



Dynamic resistance 
(ohms) 

— 



Field of view 

— 



Window material 

Sapphire 




Fig. 11-13. Detector data sheet (from NOLC Report 564). 


11.5. Theoretical Limit of Detectivity 

The optimal performance of an infrared detector would occur when the inherent 
detector noise was negligible compared to the noise induced in the detector by the 
random arrival rate of photons coming from the target. Thus far, to the author’s 
knowledge, this ultimate performance has not been realized. There are, however, 
a number of commercially available infrared detectors which under the proper operating 
conditions have a limiting noise due to the random arrival rate of photons from the 




THEORETICAL LIMIT OF DETECTIVITY 


513 


background. The background is composed of the atmosphere, spectral filter, window 
material, mirrors, and other objects besides the target in the field of view. Detectors 
which can give this type of performance include PbS, PbSe, InAs, InSb, Ge:Hg, Ge:Cu, 
Ge:Cd, Ge-Si:Au, and Ge-Si:Zn. 

11.5.1. Derivation of D* for Photon Noise Limitation. According to Planck’s law, 
the power radiated into a hemisphere per unit wavelength from a blackbody is given by 

W k = (2nc 2 h)k~ 5 (e hcl XkT - l)- 1 (11-12) 

Since there are k/he photons sec -1 w -1 at wavelength k, the number of photons sec -1 
cm -2 /x _1 can be expressed as 

n(k) = (2nc)k~ 4 (e hclXkT - l) -1 (11-13) 

where n(k) can be thought of as the average number of photons. According to the 
Bose-Einstein relation, the mean square fluctuation in the number of photons is given by 

( ghc/XkT \ 

e »c/»r _ J dl-14) 


where n is the average number of photons. Therefore the mean square fluctuation 
(in photons) of wavelength k emitted by a blackbody becomes 

--^ 277C 1 ghc/XkT 

(A " 2) x = T 7 * (e hclXkT - 1) (e hclkkT - 1) 


Since the detectors mentioned above are sensitive to wavelengths shorter than 30 /x, 
hc/k >> kT, which means that exp ( hc/kkT ) >> 1, and this permits the following 
approximation: 


(An 2 )x 


2 TTC . 

_ p-hc/XkT 

k 4 


(11-15) 


To find the total mean square photons in the spectral region to which the detector is 
sensitive, we must integrate the above expression: 


Therefore 


A n 2 ~ 2 ttc 
'kT\ 


f 


k~ 4 e~ hclKkT dk 
kT\ 2 2 kT 


(11-16) 




+ 


hck c k, 2 


The rms fluctuations in the electrical bandwidth A f and a detector area A is given by 


V2 A/ 1 An 2 At) 

where r) is the quantum efficiency of the responsive element. The generation rate 
(G x ) of carriers due to signal photons is given by 

G. = -£->}P* 

he 

where P s is the incident signal power in rms watts. For a signal-to-noise ratio of unity, 
G s must be the same as the generation rate due to the noise source, or 

ky]P s / - -- 

—-= V2 &f/kn 2 Ar) 

he 

P s is therefore the signal power necessary to produce a signal-to-noise ratio of unity, 
and is by definition the NEP. 


he / - ■■ - 

NEP = P S = — V2 A^An 2 Ar) 
kr) 
























514 


DETECTORS 


Substituting in the expression for Arc 2 one obtains the theoretical limit of NEP. 

:] 


lAnkT A f h rl2\ IcT 

NEP = 2 J --- - e~ hc,2kckT 


y]h 


kT 2 2 kT 1 12 

2 — + - + 


he hch. 


(11-17) 


Using the definition of D* from Table 11-1 and the above expression: 




e hc/2\ c kT 



2 kT 
k c hc 



(11-18) 


For this derivation a detector with a viewing solid angle of n steradians is assumed. 
If the viewed solid angle, ft, has circular symmetry, it can be expressed in terms of the 
cone angle as follows 

ft = tt sin 2 (0/2) (11-19) 

where 9 is the full cone angle. The dependence of D* upon this cone angle becomes 




e hc/2K c kT 



2 kTk c 
+ he 



( 11 - 20 ) 


From Eq. (11-20) one can observe the effects of field of view or changes in \ c upon D*. 
The effect of changing the field of view can be seen in Fig. 11-14. Here the relative 
value of D* is plotted as a function of the cone angle 6. Figure 11-15 gives D* at \ p 
as a function of long-wavelength cutoff with background temperature as a parameter. 
Equations (11-19) and (11-20) and Fig. 11-14 and 11-15 as they are apply to photo- 
emissive and photovoltaic detectors. For photoconductive detectors, which are sen¬ 
sitive to the population of carriers in the conduction band and for which the fluctuation 
in recombination rates is significant, Eq. (11-20) must be divided by V2 and the neces¬ 
sary scaling adjustment must be made on the figures. 

Similar calculations can be made for thermal detectors which are limited by the 
fluctuations in the absorbed power. However, in this case one must also consider the 



Fig. 11-14. D* and D„* as a function 
of angular field of view. 





















THEORETICAL LIMIT OF DETECTIVITY 


515 



Fig. 11-15. D„* versus long wavelength cutoff 
for background limited detection. 


contribution to the noise made by random fluctuations in the power emitted by the 
detector. The resulting equation for D* is 


4 X I0 16 e 1/2 

r\ ^ _ _ 

~ (7V + 7V) 1/2 


cm (cps) 1/2 


w 1 


( 11 - 21 ) 


where T 2 is the background temperature, T x is the detector temperature, and e is the 
emissivity of the sensitive element. Figure 11-16 presents D* for photon-noise-limited 
thermal detectors with detector temperature as a parameter. These results assume 
a 180° field of view and an infinite spectral response. When cooled spectral filters 
are used to limit the radiation striking the sensitive element to a narrow spectral 
bandwidth, the theoretical limit of D* approaches that of a photodetector sensitive 
to the same narrow spectral region. 

11.5.2. System Design Considerations. When detectors are limited by the random 
arrival rate of background photons, a number of interesting problems arise. Equation 
(11-20) indicates that decreasing the detector field of view (reducing 6) and decreasing 
the background temperature T, will lead to considerable enhancements in D*. In 
using this equation one must keep in mind that detector noise output contains (1) 
generation-recombination, (2) Johnson, and (3) 1/f noises. Also, one must recall that 
the detector resistance (for detectors which utilize changes in conductivity) is related 
to the number of background photons by R ~ 1 In. One can see then that reducing 
0 and T will reduce the photon noise, but 1 If and the lattice contribution to generation- 
recombination noise will be unaffected. However, Johnson noise (J.N.) will increase 
in the following way: 


Vj. N . « \ r ±kf~Kf' yj^ 





516 


DETECTORS 



T 2 (°K) 


Fig. 11-16. Photon-noise-limited D* of ther¬ 
mal detectors as a function of detector tempera¬ 
ture T i and background temperature 7V 


One finds then that reducing 6 to enhance D* is possible within limits set by the other 
noise sources. 

Three other effects occur which hamper the implementation of this enhanced detector 
performance. They are: an increased time constant due to reduction in background 
photon rate, the unwieldy, high value of R, and the resulting low noise from the detec¬ 
tor. The first problem sets a limit to scanning and tracking modes. The high re¬ 
sistance requires special care in designing the bias circuit and also in selecting the 
preamplifier. The reduced detector noise output places a further requirement on the 
preamplifier; i.e., it must have a lower noise level than the detector to insure detec¬ 
tor noise limited performance — a necessary condition to realize the full enhancements 
in D*. It may be necessary to cool the preamplifier to achieve this. 

In view of the above discussion, it is apparent that D* may be improved by adjusting 
certain parameters, but realizing this improvement is by no means a simple task. 

References 

1. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detection and Measurement of Infrared Radi¬ 
ation, Clarendon Press, Oxford (1957). 

2. M. Holter, et al., Fundamentals of Infrared Technology, Macmillan, N.Y. (1962). 

3. P. W. Kruse, L. D. McGlauchlin, and R. B. McQuiston, Elements of Infrared Technology, Wiley 
N.Y. (1962). 

4. W. Wolfe and T. Limperis, IRIA State of the Art Report: Infrared Quantum Detectors, 2389-50-T, 
The University of Michigan, Ann Arbor, Michigan (July 1961). 

5. H. Levinstein, Interim Report on Infrared Detectors, 105-1, Syracuse Research Institute (Feb. 
1963). 

6. Properties of Photoconductive Detectors, NOLC, Continuing series begun June 30, 1952. 

7. Interim Report on Infrared Detectors, Syracuse University, Continuing series begun Jan. 1,1954. 




REFERENCES 


517 


8. T. Limperis, "Report of the Detector Evaluation and Information Committee of the IRIS Spe¬ 
cialty Group on Detectors,” Proc. IRIS, Vol. IX, No. 1, January 1964, p. 165 (Unclassified paper 
in Secret volume). 

9. General Specifications for Environmental Testing and Associated Equipment, U.S. Army 
Specification MIL-E-5272, 13 April 1959. 

10. R. F. Potter, J. M. Pernett, and A. B. Naugle, Proc. IRE 47, 1503 (1959). 

11. P. Bratt, W. Engeler, H. Levenstein, A. MacRae, and J. Pehek, Final Report on Ge and InSb 
Infrared Detectors, Syracuse University under WADD Contract No. AF33(616)-3859 (Feb. 
1960). 

12. R. C. Jones, D. Goodwin, and G. Pullan, Standard Procedure for Testing Infrared Detectors 
and for Describing Their Performance, Office of Director of Defense Research and Engineering, 
Washington, D.C. (Sept. 1960). 

13. R. B. McQuistan, J. Opt. Soc. Am. 48, 63 (1958). 

14. M. Garbuny, T. P. Vogl, and J. R. Hansen, Rev. Sci. Inst. 28, 826 (1957). 

15. S. Nudelman and J. T. Hickmott, Bull. Am. Phys. Soc. 4, 153 (1959). 

16. A. N. Lowan and G. Blanch, J. Opt. Soc. Am. 30 (1940). 

17. K. Lark-Horovitz, V. A. Johnson, and L. Marten (eds.) Methods of Experimental Physics, 
Academic Press, N.Y. (1959). 







































































Chapter 12 

DETECTOR 
COOLING SYSTEMS* 


CONTENTS 

12.1. General. 520 

12.1.1. Limitations of Fluid Cooling. 520 

12.1.2. Cooling System Design Criteria. 522 

12.2. Types of Detector Cooling Systems. 522 

12.2.1. Direct-Contact Coolers. 525 

12.2.2. Joule-Thomson Coolers. 532 

12.2.3. Expansion-Engine Coolers. 540 

12.2.4. Thermoelectric Coolers. 544 

12.2.5. Dewars. 557 

12.3. Space-Environment Cooling Systems. 561 

12.3.1. Operating Principles. 561 

12.3.2. Design. 562 

12.3.3. Practical Cooling Systems. 563 

12.4. List of Manufacturers. 566 

^Material prepared by the technical writing staff of McGraw-Hill, Inc. 


519 


















12. Detector Cooling Systems 


12.1. General 

The proper operating temperature for an infrared detector is determined by studying 
the effect of temperature on the detector parameters, and selecting the temperature 
that provides the optimum results for the detector under consideration. Currently, 
the operating temperature of infrared detectors ranges from 1.2° K for impurity- 
activated indium antimonide, to ambient room temperature of 300° K, and above, for 
certain types of lead salts. The major requirements of a detector cooling system are: 
long operating time, stable temperatures, light weight, small size, and maximum 
reliability. Baths of liquefied gases are the most frequently used method of providing 
low temperatures because of their simplicity of design and compactness [1]. 

12.1.1. Limitations of Fluid Cooling. The temperature ranges available using 
baths of liquefied gases are shown in Fig. 12-1. In principle, a liquefied gas can provide 
constant temperature from the triple point to the critical point; however, except for 


70 


60 


50 


LEGEND 

TP - Triple Point °K (1 atm) 
BP - Boiling Point °K (1 atm) 
CP - Critical Point °K (1 atm) 


CP 154.78/50.14 CP 126.16/33.54 
BP 90.19/1 BP 77.34/1 


I 


TP 63.18/ 124 


TP 54.4/<.01 


w 

K 

H 

< 


Oh 

2 

w 


40 


30 


20 


10 


CP 44.46/26.86 



TP 13.84/.0695 


CP 3.34/1.15 
|~B BP 3.2 

0 LMi- 


CP 5.26/2.26 
IBP 4.2/1 


I 


Melting Point 3.46 
Transition Point 2.186 


Helium 3 Helium 4 Hydrogen Hydrogen Deuterium 

Deuteride 


Neon 


Oxygen 


Nitrogen 


Fig. 12-1. Temperature range of selected low-temperature liquids. 


520 




























GENERAL 


521 


special applications, the boiling point is the usual upper limit. As may be seen from 
Fig. 12-1, liquefied gases do not exist over the entire temperature range; the lowest 
temperature at which it is practical to use a liquid bath is 0.3°K, and gaps exist between 
4° and 14°K, and between 30° and 55° K. When a gas is above its critical point, it can 
be made to cool at a temperature below its inversion temperature by adjusting the gas 
flow to balance the refrigeration produced against the heat load. Temperature stability 
is sacrificed in operation above the critical point because the heat capacity of gas, as 
compared to its liquid state, is small. Variation in the heat load therefore creates 
variation in temperature. 

The temperature of the liquid bath can be varied, as shown in Fig. 12-1, by changing 
the pressure above the liquid by means of a pump and throttle valve. The stabiliza¬ 
tion of a temperature by holding constant the pressure above the liquid is directly 
related to the vaporization-temperature history of the liquid. Within the range from 
the triple point to the critical point, any desired bath temperature can be maintained 
by holding the pressure constant. At the same time the pumping rate is adjusted to 
remove precisely the amount of gas vaporized by heat leak into the bath. The cryogenic 
data for most gases are given in Table 12-1. 

Eventually, all baths of liquefied gases boil away because of heat leaks caused by the 
processes of conduction, convection, and radiation. These processes usually operate 
simultaneously; however, it is often possible to reduce to a negligible amount the con¬ 
tribution of all but that due to conduction along the solid supports, leads, piping, glass 
walls, and the insulation itself. The properties of some selected, highly efficient 
insulating materials suitable for use at low temperatures are given in Table 12-2. 


Table 12-1. Cryogenic Data [2]. 



Boiling 

Point 

1 Atm 

(°K) 

Melting 

Point 

1 Atm 

(°K) 

Liquid 
Density 
at bp 
(kg/ml 

X 10- 3 ) 

Gas 

Density 
at 273°K 
& 1 Atm 
(kg/ml 
x 10- 3 ) 

Vapor 
Density 
at bp 
(kg/ml 

x 10- 3 ) 

Vapor 
Pressure 
Solid 
at mp 
(mm) 

Heat of 
Vapor 
at bp 
(joules/ 
kg x 10- 3 ) 

Heat of 
Fusion 
at mp 
(joules/ 
kg x 10 -3 ) 

Critical 

Tempera¬ 

tures 

(°K) 

Critical 

Pressure 

(atm) 

Critical 
Volume 
(liter/kg 
X 10" 3 ) 

He 3 

3.2 

(25 atm) 

— 

— 

— 

— 

— 

— 

— 

- 

— 

He 4 

4.2 

(29 atm) 

0.125 

0.1785 

17.0 

— 

20.5 

4.183 

5.2 

2.26 

0.0144 

h 2 

20.39 

13.98 

0.071 

0.0899 

1.286 

54.0 

44.8 

58.15 

33.19 

12.98 

0.03321 

d 2 

23.6 

18.7 

0.173 

0.167 

2.58 

12.8 

286 

50 

38.3 

16.2 

0.0142 

t 2 

25.1 

21.6 

— 

— 

— 

188 

— 

- 

43.7 

20.8 

0.0089 

Ne 

27.2 

24.47 

1.2 

0.901 

9.5 

323 

87 

16.72 

44.5 

26.8 

0.002 

n 2 

77.37 

63.4 

0.808 

1.250 

4.415 

96.5 

199 

25.52 

126.1 

33.5 

0.00321 

CO 

81.6 

68 

0.812 

1.186 

— 

— 

213.5 

29.27 

133.8 

35 

0.0032 

f 2 

85.24 

53.6 

1.513 

1.71 

— 

0.1 

171.5 

13.4 

144.8 

55 

- 

A 

87.4 

83.6 

1.391 

1.78 

5.03 

516 

162.7 

28.05 

150.8 

48 

0.0019 

o 2 

90.1 

54.9 

1.14 

1.43 

4.75 

2 

212.5 

13.8 

154.1 

50.1 

0.0023 

ch 4 

111.7 

90.7 

0.425 

0.72 

1.76 

71 

581 

60.25 

190.5 

45.8 

0.008 

Kr 

120.3 

116 

2.4 

3.75 

8.33 

550 

108 

16.3 

209.3 

54.5 

- 

R 14 CF 4 

145.14 

89.5 

1.62 

— 

7.2 

— 

134.8 

- 

227.5 

37 

- 

o 3 

161.3 

80.5 

1.46 

2.14 

— 

- 

316 

13.8 

261.1 

54.6 

0.00306 

Xe 

165.3 

150.5 

3.1 

5.93 

9.77 

615 

96.25 

119.1 

290 

58 

0.00086 

C 2 H 4 

169.3 

104 

0.578 

1.19 

2.08 

— 

481 

148.5 

282.8 

50.9 

0.0045 

n 2 o 

183.6 

183 

1.23 

1.84 

— 

658 

250.5 

95 

309.7 

71.7 

0.0022 

c 2 h 6 

184.8 

90 

0.562 

1.28 

0.32 

- 

490 

96.25 

305 

48.8 

0.0048 

c 2 h 2 

189.1 

191.2 

0.623 

1.09 

- 

- 

916 

- 

309 

62 

- 

r, 3 ccif 3 

192 

91.6 

1.505 

— 

7.9 

— 

146.4 

179.9 

302 

3.9 

- 

co 2 

194.6 

215.7 

1.51 

1.87 

— 

- 

574 

71.6 

304.5 

73 

0.0022 

c 3 h« 

226.1 

77.5 

0.604 

1.78 

- 

- 

439.5 

- 

365 

45 

- 

r 22 chcif 2 

232.5 

113 

1.414 

21.3 

4.65 

- 

235 

35.15 

369 

48.7 

- 

nh 3 

239.8 

195 

0.683 

0.77 

0.898 

45 

1363 

- 

405 

111.2 

0.0042 

Ri 2 CC1 2 F 2 

243.1 

118 

1.488 

17.7 

6.25 

- 

167.2 

- 

384 

39.6 

- 

ch 3 ci 

249.4 

— 

0.993 

5.93 

2.56 

- 

427 

- 

- 

— 

— 

so 2 

263.1 

198 

0.80 

4.49 

3.2 

- 

388 

- 

430 

77.7 

0.002 

C 4 H,o 

272.5 

— 

— 

2.53 

— 

- 

- 

- 

426 

36 

— 

r„cci 3 f 

296.8 

162.7 

1.48 

2.47 

5.93 

- 

237.3 

- 

471 

43.1 

- 

c 3 h 8 

230.8 

85.9 

0.595 

1.92 

2.08 

- 

342 

79.9 

370 

42 

— 


522 


DETECTOR COOLING SYSTEMS 


Table 12-2. Physical Properties of Selected Cryogenic Insulation [2,3]. 


Insulation 

Pressure 
(mm Hg) 

Temperature 

Range 

(°K) 

Mean Thermal 
Conductivity 
(/xw cm -1 °K) 

Laminae (Cryenco) 

■'T 

1 

o 

rH 

300-77 

0.5-2 

Opacified Silica Aero Gel 

10- 2 

300-77 

2-7 

Silica Aero Gel 

IO 3 

76-20 

2 

Silica Aero Gel 

lO- 2 

300-77 

20-25 

Perlite 

lO- 2 

300-77 

6.5-11 

Perlite — 30 mesh 

760 

300-77 

330 

Perlite — 80 mesh 

<io - 3 

300-77 

10.5 

Fiber-Type Glass Fiber 

io - 3 

422 

5.8 

Heat-Felted Glass Fiber (AA Fiber) 

lO- 2 

300-77 

7.1 

Laminated Type (Linde SI-4) 

— 

300-88.5 

0.4 

NRC-I 

io - 5 

300-20 

0.9-1 

0.008-in. glass paper, 0.0023-in. 
aluminum foil — 55 shields/in. 

<10- 5 

300-20 

0.4 


12.1.2. Cooling System Design Criteria. The main variables entering into the 
design of the particular infrared cooling system are: 

1. Refrigeration load 

a. Radiation load on cooled surfaces 

b. Conduction leaks through mechanical supports 

c. Conduction leaks through electrical leads 

d. Cell bias power 

2. Operating time at rated load 

3. Standby time with no refrigerant flow 

4. Cooling temperature and tolerances 

5. Environmental operating conditions 

6. Weight and space requirements 

7. Detector cell configuration and dimensions 

8. Length and type of feed lines (especially critical in liquid transfer coolers) 

Once this information has been obtained, a choice can be made as to the type and 
the capacity of the cooling system. 

12.2. Types of Detector Cooling Systems 

The five types of detector cooling systems are: direct contact, Joule-Thomson, ex¬ 
pansion-engine, thermoelectric, and magneto-thermodynamic. Of the five, three are 
primarily mechanical, one is electrical, and one is magnetic. The three mechanical 
cooling systems can be further classified as open or closed cycle, depending on whether 
the evaporated coolant is vented to the surroundings or recycled. Magneto-thermo¬ 
dynamic cooling, although a promising process for the generation of temperatures 
approaching a few microdegrees Kelvin, is not currently used for detector cooling 
because the state of the art is such that constant low temperatures cannot be maintained 
for any length of time. Table 12-3 lists the characteristics of the various types of 


TYPES OF DETECTOR COOLING SYSTEMS 


523 


Table 

12-3. Infrared Detector Cooling Systems Characteristics 


Cooler 

Type 

Temperature 

Range 

(°K) 

Cooling Capacity 
(watts) 

Power Input 
(watts) 

Cooldown 

Time 

(min) 

Direct Contact 

Integral 

4.2-77 

0.05-10 

None 1 

0-15 


Liquid feed 

4.2-77 

0.05-10 

None 1 

0.05-15 

Joule-Thomson 

Single stage, 
open cycle 

20-80 

0.01-5 

None 1 

1-5 


Multiple stage, 
open cycle 

4.2-27 

5-20 

None 1 

5-20 


Single or multistage 
closed cycle 

4.2-77 

1.0-30 

250-1000 

5-30 

Expansion 

Engine 

Piston-Regenerator 

30-300 

0.01-1.0 

400-800 

3-10 


Displacer- 

Regenerator 

30-300 

1-10 

200-300 

5-10 


Turbine 

30-300 

1.5-5 

400-600 

5-10 

Thermoelectric 

Single stage 

250-300 3 

0.01-0.2 

0.5-3 

2 


Cascaded multiple 
stages 

195-250 3 

0.01-0.02 

1-5 

2 


524 


DETECTOR COOLING SYSTEMS 


Table 12-3. Infrared Detector Cooling Systems Characteristics ( Continued ) 


Cooler 


Design Features 


Design Limitations 


Direct Contact 


Excellent reliability, light weight, Short operating time limited by liquid 
small bulk, low pressures 2 capacity; standby time limited by evap¬ 

oration rate 


Excellent reliability, remote cooling, Short operating time limited by liquid 
light weight, small bulk, simplified capacity; standby time limited by evap- 
installation problems, low pressures 2 oration rate, losses in transfer lines 


Joule-Thomson Remote cooling, light and small cool¬ 
ing head, simplified installation 
problems 2 


Low temperatures obtained, remote 
cooling, light and small cooling 
head, simplified installation prob¬ 
lems 3 


Continuous duty, long operating 
time, low temperatures obtained, 
remote cooling, light and small cool¬ 
ing head, simplified installation 
problems 

Expansion Continuous duty, long operating 

Engine time, low pressures 


Easily clogged; requires high pressures 
and flow rates; high leakage and con¬ 
tamination; short operating time lim¬ 
ited by tank capacity; only portion of 
tank used because of pressure drop 

Needs precooling with separate gas cycle; 
easily clogged; requires high pressures 
and flow rates; high leakage and con¬ 
tamination; short operating time lim¬ 
ited by tank capacity; only portion of 
tank used because of pressure drop 

Requires highly loaded, high-pressure, 
noncontaminating compressor; poor life 
and nonreliable; needs precooling for 
low temperatures; easily clogged; high 
leakage and contamination 

Requires noncontaminating compressor; 
great wear and constant friction in ex¬ 
pander; high leakage around expander 
valves; low regenerator efficiency, mi¬ 
crophonics 


No valves required, continuous duty, Great wear and constant friction between 
long operating time, low pressures, piston, displacer, and cylinder; poor 
low friction with gas bearings compression; low regenerator efficiency; 

microphonics 


Thermoelectric 


No valves required, continuous duty, 
long operating time, low pressures, 
low friction with gas bearings 

Excellent reliability, static operation, 
light weight, very small, long life 

Excellent reliability, static operation, 
light weight, very small, long life, 
lower temperatures 


Requires high flow rate, means of re¬ 
moving work done on turbine, separate 
compressor; strong mechanical construc¬ 
tion required because of high speed 

Relatively high-temperature operation; 
low cooling capacity; hard to obtain 
operating voltages and currents 

Relatively high-temperature operation, 
very low cooling capacity 


'May require 20 to 100 w for storage-tank heater and for control system. 

2 Size and weight of complete system depends on storage-tank capacity and operating time. 

3 Lower temperatures will be available when more efficient thermoelectric materials under investigation are 
developed. 


TYPES OF DETECTOR COOLING SYSTEMS 


525 


detector cooling systems presently available commercially. Manufacturers of detector 
coolers are listed at the end of this chapter. The temperature ranges of these systems 
are shown in Fig. 12-2. For the mechanical cooling systems, the temperature range 
is dependent on the fluid used. The range of the common liquids is shown; however, 
other fluids and custom-built designs can broaden these limits. 


Fahrenheit 

(F) 


Centigrade 

(C) 


Kelvin 

(K) 


Boiling Point, 
Water 


Freezing Point, 
Water 


Absolute Zero 


250 

212 

200 

150 

100 

50 

32 

0 

-50 

-100 

-150 

-200 

-250 

-300 

-350 

-400 

-450 

-460 


100 

80 

60 

40 

20 


-40 

-60 

-80 

-100 

-120 

-140 

-160 

-180 

-200 

-220 

-240 

-260 

-273- 


380 

•373 

360 

340 

320 

300 

280 

273 

260 

240 

220 

200 

180 

160 

140 

120 

100 

80 

60 

40 

20 

— 0 



Single-Stage 

Thermoelectric 


Cascaded 

Thermoelectric 


Expansion Engine 


Joule-Thomson 


J 

Liquid Transfer 


Fig. 12-2. Temperature range of commercially available infrared detector 
cooling systems. 


12.2.1. Direct-Contact Coolers. There are two types of direct-contact coolers: 
integral and liquid feed. The integral cooler is the most common of all cooling systems. 
In its simplest form, it consists of nothing more than a detector dewar into which 
the coolant is poured. The liquid-feed cooler is a more involved version of the basic 
direct-contact cooler. The coolant liquid is fed from a liquid storage tank through 
transfer lines to the cooling head. The principal advantage of the liquid-feed cooler 
is its installation flexibility; the cooling head can be mounted remote from the liquid- 
storage container. Table 12-4 lists specific models of commercially available direct- 
contact coolers. 

















526 


DETECTOR COOLING SYSTEMS 
Table 12-4. Liquid-Transfer Coolers. 


Manufacturer 

Model 

Type 

Coolant 

Capacity 

(liters) 

Operating 

Temperature 

(°K) 

Cooldown 

Time 

(min) 

Ai Research 

134338-1 

Liquid feed 

n 2 

5.0 

77 

a 


134642 



n 2 

1.5 

77 

3 


134548 




1.0 


3 

ITT 

— 

Liquid feed 

n 2 

1.83 

77 

<2 


Linde 


LNI-1 

LNI-13 

LNI-18 

LNI-3 

LNI-4 

LNI-5 

LNI-12 

LNI-15 

LNI-28 

LNI-9 

LNF-2 


Integral 


Liquid feed 


N 2 


Ne 

Ne/He 

N 2 

Ne/He 

N 2 

n 2 

n 2 


0.174 

0.282 

0.198 

0.902 

0.174 

0.209 

0.777 

0.395 

0.062 

1.33 

0.450 

1.181 

1.524 


77 

27 

Ne=27 

He=4 

77 


a 


LNF-3 

LNF-4 


0.456 

3.159 77-166 


LNF-5 

LNF-6 

LNF-12 

LNF-13 


3.159 77 

0.519 
0.405 
1.128 


Raytheon 

QKN 748 & 1003 

Integral/liquid feed 

n 2 

0.00005 

77 

a 


QKN 884 & 1004 

i 



0.00119 

i 




QKN 1204 

Integral 



0.045 

65-77 




QKN 1205 


> 

t 

0.007 

i 




QKN 1206, 1207 

Integral/liquid feed 

Ne/He 

0.05/0.5 

Ne=27 




& 1208 


n 2 


He=4 




Storage dewar 

1 

i 


i 




Transfer 

Liquid feed 

n 2 

0.147 

77 


2 


Pumped transfer 

1 

i 


68 


2 

SBRC 

LNI-E 

Liquid feed 

n 2 

1.5 

77 

<1 






also 1 & 3 




Hughes 

DP-099 

Liquid feed 

n 2 

1.356 

77 

<3 

Aircraft Co. 










DP-001 

Liquid feed 

n 2 

1.163 

77 

<3 


AP-111 

Liquid feed 

n 2 

3.172 

77 

<3 























TYPES OF DETECTOR COOLING SYSTEMS 
Table 12-4. Liquid-Transfer Coolers ( Continued ). 


527 


Standby 

Time 

(hr) 

Evaporation 

Rate 

(kg/hr) 

Operating 

Time 

(hr) 

Method of Filling 

Length 

(m) 

Diameter 

(m) 

Weight 

Empty 

(kg) 

Full 

(kg) 

24 

0.0081 

30-50 

Pressure lines 

0.254 

0.28 

3.4 

7.5 

24 

0.0136 

6 



0.406 

0.127 

1.59 

2.94 

24 

0.00906 

3.5 



0.178 

0.1525 

2.27 

3.08 

83 

0.0181 

0.3 

Pour or pressure 

0.292 

0.1568 

2.72 

4.2 


8.5 

0.0168 

8.5 

Pour 

0.1015 

0.076 

0.202 

0.344 

8.0 

0.0281 

8.0 

Insulated fill lines 

0.146 

0.1175 

1.81 

2.02 

8.0 

0.0199 

8.0 

Pour 

0.133 

0.0761 

0.226 

0.386 

29.0 

0.025 

29.0 



0.33 

0.089 

0.78 

1.455 

10.0 

0.014 

10.0 



0.1365 

0.0794 

0.163 

0.304 

9.0 

0.0186 

9.0 



0.2255 

0.0761 

0.286 

0.454 

8.0 

0.0785 

8.0 

Insulated fill lines 

0.203 

0.1175 

2.04 

2.66 

Ne=100 r 

Ne=0.00454 

Ne= 

=100“ 

Insulated fill lines, 

0.277 

0.089 

0.955 

d 

He=3“ 

He=0.0168 

He= 

=3“ 

pour for shield 






Ne=190“ 

Ne=0.0081 

Ne= 

= 190“ 



0.394 

0.127 

1.585 



He=5.75“ 

He=0.029 

He= 

=5.75“ 








2.0 

0.476 

2.0 

Pour 

0.254 

0.1142 

1.61 

2.6 

24.CP 

0.0172 


h 

Pressure 

0.464 

0.1142 

1.36 

2.58 

24.0" 

0.0059 



Pour or pressure 

0.254 

0.089 

0.905 

1.275 

204 

0.0122 



Pour 

0.222x0.1715 

_ 

3.17 

5.61 







X0.343 





204 

0.0122 





0.343 

0.1525 

1.27 

3.81 

<1 

0.666 





0.305 

0.089 

0.294 

0.712 

<1 

0.870 





0.306 

0.089 

0.408 

0.735 

24 

0.02675 

3 



0.254 

0.1015 

1.585 

2.5 



48' 

48' 

0.2175' 

0.277' 

0.75 

1.5 

8 

8 

5' 

4' 

Pc 

Insulated 
pour for 
Pour or 
Pressu 

ur 

fill lines, 
shield 
pressure 
re lines 

0.0444 

0.0761 

0.0761 

0.0761 

0.4165 

0.353 

0.508 

0.711 

0.0127 

0.0127 

0.019 

0.019 

0.1142 

0.1142 

0.1525 

0.203 

0.011 

0.020 

0.0113 

0.0113 

3.4 

2.78 

2.26 

5.9 

3.74 

3.12 

3.44 

7.08 

60 


8> 

Pressure lines 

0.4835 

0.127 

2.04 


24 

0.014 

4-i 

Insulated fill lines 

0.28 

0.135 

2.04 

3.18 

24 

0.018 

4 J 

Insulated fill lines 

0.31 height 

3.22 

4.22 






0.29 width 








0.09 depth 



24 

0.028 

8 

Insulated fill lines 

0.42 

0.153 

3.98 

6.58 


“Cooled concurrently with filling. 

"Depends on size and construction of cooling head. 
c Based on a 0.09-w detector and electrical lead heat load. 
d Sum of empty weight, weight of shield liquid, and 
weight of inner cell liquid. 

“Based on 0.02-w detector and electrical lead heat load. 


•''Evaporation rate of 0.408 kg/day, 0.816 kg of liquid re¬ 
maining after 24-hr standby. 

"Evaporation rate of 0.1405 kg/day, 0.266 kg of liquid 
remaining after 24-hr standby. 

"Dependent on flow rate and heat load. 

'Based on the use of a QKN 1004 dewar. 

J Based on a 1/2-w heat load. 


















528 


DETECTOR COOLING SYSTEMS 


Table 12-4. Liquid-Transfer Coolers ( Continued ). 


Manufacturer 

Detector 

Location 

Operating Attitude 

Remarks 

AiResearch 

External 

Vertical 

Flow controlled by heat leak pressure buildup. 




Horizontal 

Flow controlled by heat leak pressure buildup. 




Vertical 

Flow controlled by heat leak pressure buildup. 

ITT 

External 

Vertical 

Flow controlled by heat leak pressure buildup; transfer- 





head cooling tube 0.419 m long, 0.00635 m i.d., 
weight 0.0396 kg. 

Linde 

Side 


Vertical 

Separate fill and vent ports. 


Bottom 

Vertical 

Separate fill and vent valves. 


End 


Horizontal 

Combination fill and vent port. 


Tail, side 

Horizontal 

Separate fill and vent ports; 0.1208-m tail. 


Tail, side 

Horizontal 

Combination fill and vent port; 0.0635-m tail. 


Tail, bottom 

Vertical 

Separate fill and vent ports; 0.092-m tail. 


Bottom 

Vertical 

Temperature is pressure dependent; cools mosaic 





detectors. 


Bottom 

Vertical 

Double cell for Ne or He with N 2 shield. 


Bottom 

Vertical 

Double cell for Ne or He with N 2 shield. 


End 


Horizontal 

Single cell; used to cool vidicon tubes. 


External 

Horizontal 

Flow controlled by electric heater pressure buildup; 





requires 24-v dc, 115 w. 




Vertical 

Flow controlled by orifice valve. 




Vertical 

Flow controlled by bridge circuit and valve regulating 





pressure buildup. 




Vertical 

Flow controlled by valve on dewar vent or temperature 
control pane'. 




<60° from vertical 

Flow controlled by evaporation rate of liquid in cell 





dewar. 




<60° from vertical 

Flow controlled by evaporation rate of liquid in cell 





dewar. 




Vertical 

Orifice-control or temperature-control devices to 





regulate flow rate, absolute or gauge pressure-relief 
valves, with optional pressure buildup heaters for 
system pressure regulation. 


Raytheon 


Bottom 


Horizontal to vertical Small metal dewar with integral detector. 


< 55° from vertical 


Double cell for Ne or He with N 2 shield. 


<55° from vertical 
Horizontal 
Horizontal 


SBRC External Horizontal 


Hughes External Horizontal 

Aircraft Co. 

External Horizontal 


External Horizontal 


Double cell for Ne or He with N 2 shield. 

Flow controlled by orifice valve. 

Flow controlled by orifice valve and pump rate. 


Flow controlled by orifice valve; 0.00519-m or 0.00828- 
m diameter cooling head; weight 0.0142 kg. 


Flow controlled by temperature-control device, relief 
valve, and heater. 

Flow controlled by orifice, relief valve, and heater. 


Flow controlled by absolute pressure relief valve and 
temperature-control device; system designed for high 
ambient temperature environment. 















TYPES OF DETECTOR COOLING SYSTEMS 


529 


12.2.1.1. Integral Coolers. The integral cooler consists of a detector in direct thermal 
contact with a supply of liquid coolant (Fig. 12-3). The detector is integrally mounted 
in a dewar that serves both as the detector mount and the liquid container. When 
a solid coolant such as dry ice is used, sticks of the coolant are inserted into the coolant 
well. Thermal contact between the solid C0 2 and the walls of the coolant well is 
ensured by mixing the solid with a low-freezing-point liquid such as acetone. A 
basic limitation of the direct-contact cooler is its operating attitude. In order to 
keep the coolant in direct contact with the detector, the dewar must be maintained 
in an essentially vertical position. For airborne and tracking instrument applications 
where the detector is moved through 360° of an arc in both horizontal and vertical 
planes, thermal contact between the coolant and the detector is maintained regardless 
of the dewar attitude. This is accomplished by using copper conducting plates that 
are spring loaded so that they remain in contact with the coolant. 


Infrared Transparent Window 



Detector Element 

Copper Mounting 
Block 

Glass Wall 

Copper Conducting 
Shield 

Coolant Well 


Fig. 12-3. Direct-contact cooler. 


12.2.1.2. Liquid-Feed Coolers. The liquid-feed cooler consists of an insulated liquid- 
storage container, transfer lines, a cooling head, and the necessary controls (Fig. 12-4). 
The transfer mechanism is either gravity or gas pressure. The gas pressure to force 
the liquid from the storage container to the cooling head originates from the natural 
pressure build-up due to thermal leakage into the storage container, or from the residual 
pressure of the filling operation. In cases where the natural pressure build-up is not 
sufficient, or better regulation is required, a small pressure-regulated electrical heater 
is placed in the storage container to evaporate the required amount of liquid (see 
Fig. 12-5). The flow of fluid is usually self-limiting to provide operation over a wide 
range of differential pressures, using only on-off control, and to prevent flooding of the 
cooling head. As the detector cell cools, the liquid in the cell evaporates and vents 
through the adjustable orifice-flow control valve, which sets the pressure differential 
between the tank and the cooling head to a value that maintains a constant rate of flow 



























530 


DETECTOR COOLING SYSTEMS 


of liquid into the cell. The pressure-control relief valve regulates the pressure exerted 
on the liquid, and also acts as a relief valve to vent the storage container in case of 
malfunction. With the addition of a small gas liquefier to the low-pressure cell vent 
line, the liquid-feed cooler becomes a closed-cycle system capable of long standby and 
operating times. 


Vent Valve Liquid Transfer Line 



Fill Valve 


Fig. 12-4. Liquid-feed cooler. 



Liquid Nitrogen Charge 
Output Pressure 
Heater Off Pressure 
Pressure Buildup Time 


Specifications 

5.0 liters Operating Time 

2000 psi 
2300 psi min 

10 min to 1500 psi Weight 

Dimensions 


5.5 hr at 5 1pm after 2 hr 
standby, 2.25 hr at 12 
lpm at 2000 psi 
22.125 lb uncharged 
14.812 x 12 x 13.5 in. 


Fig. 12-5. Barnes nitrogen pressure generator diagram [4). 


































































TYPES OF DETECTOR COOLING SYSTEMS 


531 


The usual mechanism of liquid transfer is a two-phase flow known as Leidenfrost 
transfer. When small quantities of a low-temperature liquid are passed through a 
warm tube, some of the liquid evaporates to form a gaseous skin that keeps droplets 
of the liquid insulated from the walls of the tube sufficiently well so that small quantities 
of the liquid can be efficiently transferred. The loss of liquid in the transfer process 
increases with transfer line length, typically 75% for a line 1 ft long. Figure 12-6 
shows a plot of transfer efficiency versus cooling load for several lengths of uninsulated 
transfer lines. Graphs of total system weight versus total operating time and versus 
required liquid capacity for a typical liquid-nitrogen detector cooling system are shown 
in Fig. 12-7. Using these graphs, a single-search detector load of 1 w operating for 
100 hr would require a system weighing 7.6 lb with a liquid-storage container of 5.9- 
liter capacity. 



Fig. 12-6. Transfer efficiency for liquid N 2 through uninsulated 
tubes [1]. 


DETECTOR HEAT LOAD (w) 



3 ---------- 

0 20 40 60 80 100 

OPERATING TIME (hr) 



LIQUID N 2 CAPACITY (liter) 


Fig. 12-7. System dry weight versus operating time and storage tank 
capacity [5], 

































































532 


DETECTOR COOLING SYSTEMS 


12.2.2. Joule-Thomson Coolers. The Joule-Thomson or cryostat cooler is based 
on the Joule-Thomson effect of cooling caused by the adiabatic expansion through an 
orifice of a gas that is initially below its saturation temperature. The expanded gas, 
thus cooled, is passed back over the incoming gas to cool that, which results in re¬ 
generative cooling. The process continues until liquid begins to form at the orifice to 
produce a bath of liquid at the boiling temperature of the gas. The Joule-Thomson 
cooler consists of a finned tube in the form of a coil, an orifice, and orifice cap, and an 
outer shield or coil (Fig. 12-8). The finned tube is made of very small inside-diameter 


Finned Tube Shield 

Orifice Cap \ / Inlet 

life 

Orifice 


Outlet 

Fig. 12-8. Construction of Joule-Thomson cooler. 



tubing to provide the large ratio of surface area to volume necessary for effective heat 
exchange. Dust particles and all traces of higher-freezing-point gases must be excluded 
from the gas entering the cooler; otherwise the finned tube will clog or freeze up after 
a short period of operation. The orifice is usually surrounded by an orifice cap that 
returns the expanded gases back along the finned tube. This cap must be maintained 
in close thermal contact with the dewar or detector to ensure efficient heat transfer. 
The outer shield is formed in a number of ways. A separate return tube can be coiled 
around the finned tube, the finned tube can be placed in another length of tubing of 
greater diameter, or the fined tube can be placed in a tight-fitting container so that 
the expanded gas must flow through the channels formed by the space between adjacent 
turns of the finned tubing. With the addition of a small noncontaminating compressor, 
the expanded gas can be recycled to provide continuous operation for long periods of 
time without the need for gas replenishment. Table 12-5 lists specific models of 
commercially available Joule-Thomson coolers. 

12.2.2.1. Gas-Supply Coolers. In an open-cycle Joule-Thomson cooler, bottled gas 
is the usual source of supply, although high-pressure gas containers are frequently 
used. A high-pressure regulator is required to maintain the gas pressure at approxi¬ 
mately 1500 psi since both the flow rate and cooldown time are pressure dependent 
(Fig. 12-9). Most pressure regulators that cover the range of 1000 to 3000 psi, and 



8 

6 

4 

2 

0 


c 
■ <—< 

s, 

w 

2 

l-H 

H 

O 

Z 

HH 

H 

K 

< 

CQ 


Fig. 12-9. Flow rate required to liquefy nitrogen and cooldown time as a func¬ 
tion of gas pressure for SBRC P/N 9114 cryostat in style No. 120-1 detector. 
Total detector flask heat load of 0.23 w and ambient temperature of 295° K [6]. 



















TYPES OF DETECTOR COOLING SYSTEMS 


533 


accommodate normal flow rates up to 10 liters per minute (standard temperature 
pressure), can be used. The pressure regulator must be capable of reducing the pres¬ 
sure to counteract the tendency of the flow to increase, sometimes reaching a value 
twice as much as the initial flow, as the temperature drops. Commercially available 
high-purity gas with a low water content is required for operation in the cooler. To 
ensure that all undesirable particles are removed, and that the water content is suffi¬ 
ciently low, a dryer and a filter can be inserted in the high-pressure line ahead of the 
finned tube. A chemical dryer containing an absorbent material removes the excess 
moisture; it is followed by a porous or a sintered metal filter to remove any floating 
particles. To obtain even greater gas purity, a cold trap consisting of a coiled tube 
immersed in a bath of dry ice and acetone or liquid nitrogen is inserted after the dryer 
and filter (Fig. 12-10). In operation, water vapor and gases, having a freezing point 
above that of the coolant bath, will condense on the walls of the tubing at a rate de¬ 
pendent on the initial purity of the gas being dried. 



Fig. 12-10. High-pressure cold trap with metal filter [7]. 
(Copyright ITT, Farnsworth Division). 


12.2.2.2. Liquid-Supply Coolers. The simplest way to obtain high-purity dry gas 
is to feed the Joule-Thomson cooler from a storage container of liquid coolant. A 
major problem of the liquid supply is the evaporation of liquid during standby; how¬ 
ever, this supply is still lighter and smaller than a compressed gas supply of the same 
capacity. A diagram of a typical liquid supply or nitrogen pressure generator is 
shown in Fig. 12-5. The pressure-regulated heater inside the storage tank evap¬ 
orates the liquid to provide the proper gas pressure. A cold trap, a filter, and a po¬ 
rous metal plug remove undesirable particles and all traces of higher-freezing-point 
gases. 



534 


DETECTOR COOLING SYSTEMS 


Table 12-5. Joule-Thomson Coolers 


Manufacturer 

Model 

Cycle 

Coolant 

Operating 

Pressure 

(atm) 

Flow 

Rate 

(i/min) 

Op. 

Temp. 

(°K) 

Cooling 

Capacity 

(w) 

AiResearch 

Joule-Thomson 

Closed 

n 2 

102.1 


77 

1 

Air Products 

AC-2-109 A 

Open 

H 2 ° 

122.4 

10.5 

20-75 

0.75 

Inc. 



n 2 

152.9 

9.9 




AC-2-109B 

Open 

h 2 

105.5 

9.1 

20-75 

1.5 




n 2 

122.4 

27.2 




AC-2-109C 

Open 

h 2 

102.1 

29.7 

20-75 

4.0 




n 2 

142.7 

31.2 




AC-2-109D 

Open 

h 2 

102.1 

21.0 

20-75 

6.0 




n 2 

136.0 

36.2 




NOTS-Side- 

Open 6 

n 2 

81.7- 

5.5 

80 

5-8 


winder 1C 



170.3 





IOR-OON 

Closed 

n 2 

68.0 

8.5 

80 

1 


2C 

Closed 

n 2 

54.4- 

8.5- 

80-100 

1-5 





81.7 

22.6 




Two-fluid 

Closed 

h 2 

34.0- 

17-31 

30 

0.5-1 


cascade 


n 2 

81.7 




ITT 

600 e 

Open 

n 2 

115.6 

2-6 min 

77 

0.01 





149.5 








max 





600" 

Open 

n 2 

115.6 


77 

0.01 





149.5 








max 





“Available for operation with other gases and different pressure and flow rates. 

6 Similar Joule-Thomson coolers for use with several of the gases, argon, neon, hydrogen, and helium are available. 
'Compressed N 2 supply consisting of 8.2-liter tank, solenoid-operated valve, filler, pressure gauge, and quick 
disconnect available. Operates for 1-1/2 hours, gas bottle charging cart available. Cryostats recommended for 
use with an ITT detector cell dewar. 














TYPES OF DETECTOR COOLING SYSTEMS 535 


Table 12-5. Joule-Thomson Coolers ( Continued) 


Cooldown 

Construction 

Heat Exchanger 


Required 

Accessories 

Time 

(min) 

Length 

(m) 

Diameter 

(m) 

Weight 

(kg) 

Compressor 


Bundle-type 

counterflow 

0.203 

0.089 

0.454 

Two stage, nonlubricat- 
ed piston-type, ac mo¬ 
tor driven, 115 v, 130 
w, 60 cps, 1 <f>, 0.432 x 
0.178 m diameter, 9.07 
kg. 


5.5 

Two-stage 
cascade, 

0.114 

0.00635 

0.454 

Not Used 

g 

19 

concurrent 

flow 

0.159 

0.00955 

0.454 

Not Used 

g 

12 

Two-stage 

cascade, 

0.21 

0.0127 

0.454 

Not Used 

g 

22 

concurrent 

flow 

0.33 

0.019 

0.454 

Not Used 

g 

0.67-1 

One stage 
concurrent 
flow 

0.0457 

0.00508 

0.454 

Not Used 

g 

30 

One stage, 
finned 

coiled tubing 

0.0762 

0.00635 


Two stage, nonlubricat- 
ed piston-type, ac mo¬ 
tor driven, 110 v, 250 
w, 60 cps, 1 <f>; or 208 v, 
600 w, 400 cps, 3 <£, 
0.127 x 0.205 x 0.305 
m, 7.25 kg. r 

g 

10-25 

One stage, 
finned 

coiled tubing 

0.0762- 

0.152 

0.00635- 

0.0127 


Two stage, nonlubricat- 
ed piston-type, ac mo¬ 
tor driven, 110 v, 300- 
1000 w, 60 cps, 1 </>; or 
208 v, 300-1000 w, 400 
cps, 3 0, 0.127 x 0.205 
x 0.28-0.381 m, d 7.25- 
12.7 kg.° 

g 

20-30 

Two stage, 
finned 

coiled tubing 

0.152- 

0.239 

0.00955- 

0.0127 


Two nonlubricated, two 
stage piston-type, ac 
motor driven, 110 v, 
800-2000 w, 400 cps, 

3 <b\ or 208 v, 800-2000 
w, 400 cps, 3 <t>, 0.127 x 
0.205 x 0.28-0.381 m, c 
15.9-22.8 kg. c 

g 

20 

One stage, 
coiled tubing 

0.114 

0.00955 


Not Used 

g 

20 

One stage, 
coiled tubing 

0.143 

0.00476 


Not Used 

g 


C A 2-stage, 6-cylinder compressor has been developed for use where lower temperatures and greater cooling capacity 
are required, but it is not recommended for infrared use. 

"The 2-fluid cascade cooler requires two model 2C units, each with the listed dimensions, which can be packaged 
together or separately as desired. 

’Filter absorber system is a required accessory. Contact firm for details. 















536 


DETECTOR COOLING SYSTEMS 


Table 12-5. Joule-Thomson Coolers ( Continued ) 


Manufacturer 

Model 

Cycle 

Coolant 

Operating 

Pressure 

(atm) 

Flow 

Rate 

(i/min) 

Op. 

Temp. 

(°K) 

Cooling 

Capacity 

(w) ' 

Honeywell 


Open or 

n 2 

136.0 

3 start 

77 

1 



closed 


102.1 

4.5 steady 



Hughes 

JTC-001 

Closed 

n 2 

68.0- 

9 each for 

77 

0.4 for 

Aircraft Co. 




103.6 

6 J-T 


each J-T 






heat 








exchr 




JTC-002 

Closed 

A 

68.0- 

7.5 each for 

87 

0.4 for 





103.6 

6 J-T 


each J-T 






heat 








exchr 




JTO-OOl 

Open 

A 

406.5- 

40-10 

87 

40-10 





116.1 





JTO-002 

Open 

A 

358.1- 

15-8 

87 

10-1 





62.9 




Philco 

C100 

Open 

n 2 

68.0- 

3 start 

77 






204.2 

9 steady 




C150 


n 2 



77 


Raytheon 

QKN 748/1003 

Open 

n 2 



77 



QKN 884/1004 

Open 

n 2 

95.2 

31 




QKN 961/1005/ 

Open 

H 2 /Ne 



20/28 



1007 


n 2 



65-78 



QKN 1204 

Open 

n 2 



65-78 



QKN 1205 

Open 

n 2 



65-78 
















TYPES OF DETECTOR COOLING SYSTEMS 


537 


Table 12-5. Joule-Thomson Coolers ( Continued) 


Cooldown 


Heat Exchanger 


Required 

Accessories 

Time 

(min) 

Construction 

Length 

(m) 

Diameter 

(m) 

Weight 

(kg) 

Compressor 

2 

One stage, 
coiled tubing 

0.122 

0.0042 

min 


Not Used 

g 

4.5 

One stage, 

0.051 

0.0083 

0.030 

Four stage, nonlubricat- 



finned 

coiled tubing 




ed, piston type with 
unique seal separating 
lubricated and nonlu- 


3.0 

One stage, 




bricated areas, ac mo- 



finned 




tor driven, 208 v, 400 



coiled tubing 




cps, 3 <)>, 1300 w input; 

2 compressors with 
common drive can be 
driven hydraulically or 
electrically; both di¬ 
rect and belt drive; 
0.41 x 0.46 X 0.28 m, 
30 kg. 


0.15 

One stage, 

0.032 

0.005 

0.010 

Not Used 

h 


integrally 

finned 

tubular 






0.5 

exchanger 

One stage, 

0.051 

0.0063 

0.015 

Not Used 

h,i 


integrally 

finned 

tubular 

exchanger 






1-1.5 

One stage, 

0.07 

0.0097 

0.085 

Not Used 

g 


coiled tubing 





0.75-0.92 

One stage, 
coiled tubing 

0.07 

0.00524 

0.085 

Not Used 

g 


One stage, 

0.0317 

0.00515 


Not Used 

g 


finned 

coiled tubing 

0.0635 

0.00515 


Not Used 

g 


Two stage, 

0.2135 

0.00896 


Not Used 

g 


coiled tubing 
One stage, 

0.0635 

0.00896 


Not Used 

g 


finned 

coiled tubing 

0.0635 

0.00514 


Not Used 

g 




min 






0.0127 







min 





9 Filter absorber system is a required accessory. Contact firm for details. 
h Storage tank required. 

'Manifold required. 















538 


DETECTOR COOLING SYSTEMS 


Table 12-5. Joule-Thomson Coolers ( Continued ) 


Manufacturer 

Model 

Cycle 

Coolant 

Operating 

Pressure 

(atm) 

Flow 

Rate 

(£/min) 

Op. 

Temp. 

(°K) 

Cooling 

Capacity 

(w) 

SBRC 

9114 

Open 

Freon 

17.0 


192 

0.2-1 




A 

68.0 


80 





n 2 

81.7 

4.5 

77 



9174 

Open 

Freon 

17.0 


192 

0.2-1 




A 

68.0 


80 





n 2 

81.7 

4.5 

77 



9186 

Open 

Freon 

17.0 


192 

0.2-1 




A 

68.0 


80 





N 

81.7 

4.5 

77 



9185 

Closed 

Freon 

17.0 


192 

0.2-1 




A 

68.0 


80 





n 2 

81.7 

4.5 

77 



80°K —0.5 w 

Closed 

n 2 



82 

2 


80 K 5 w 

Closed 

n 2 

136.1 


80 

5 


80 K 2 w 

Closed 

n 2 

102.1 


80 

2 


82 K .5 w 

Closed 

n 2 

102.1 


84 

1 

Stratos 

S-2108 

Closed 

n 2 

199.8 

9.06 

26 

1.0 




h 2 

136.0 

7.08 



Westinghouse 


Closed 

n 2 

102.1 

7.08 

90 

0.5 



Closed • r 

Ne/N 2 



43 



/2-liquid closed-loop cooler and dual-diaphragm compressor under development. 














TYPES OF DETECTOR COOLING SYSTEMS 539 


Table 12-5. Joule-Thomson Coolers ( Continued ) 


Cooldown 


Heat Exchanger 


Required 

Acessories 

Time 

(min) 

Construction 

Length 

(m) 

Diameter 

(m) 

Weight 

(kg) 

Compressor 

<3 

One stage, 
finned 

coiled tubing 

0.042 

0.00519 

0.00227 

Not Used 

8 

<3 

One stage, 
finned 

coiled tubing 

0.0418 

0.0083 

0.00255 

Not Used 

8 

<4 

One stage, 
finned 

coiled tubing 

0.0508 

0.00519 

0.00227 

Not Used 

8 

<4 

One stage, 
finned 

coiled tubing 

0.0478 

0.0083 

0.00255 

Not Used 

8 

5 

One stage, 
finned 

coiled tubing 

0.0508 

0.00519 

0.02834 

4 cylinder piston, ac mo¬ 
tor driven, 208 v, 225 
w, 400 cps, 3 4 >, 0.241 x 
0.152 m diameter, 4.54 
kg. 

3-stage piston, ac motor 
driven, 208 v, 525 w, 
400 cps, 3 4>> 3.175 x 
1.78 m diameter, 7.3 
kg. Air cooled with 
own fan. 

3-stage piston, ac motor 
driven, 208 v, 325 w, 
400 cps, 3 4> , 3.175 x 
1.78 m diameter, 7.3 
kg. Air cooled with 
own fan. 

3-stage piston, ac motor 
driven, 208 v, 190 w, 
400 cps, 3 </>, 2.41 x 
1.524 m diameter, 4.55 
kg. Air cooled with 
own fan. 



Two stage, 
cross flow 
tubing 

0.152 

0.0381 

0.136 

Two cycle, four stage, oil 
activated metallic dia¬ 
phragm, ac/dc, hydrau¬ 
lic or air turbine driv¬ 
en, 0.208 x 0.216 m 
diameter. 

j 

15 

One stage, 
coiled tubing 

Two stage, 
coiled tubing 

0.0381 

0.0762 

0.00955 

0.0127 


Oil activated, metallic 
diaphragm, dc motor 
driven, 28 v, 1 w, 0.165 
x 0.165 x 0.127 m, 
4.54 kg. 

Oil activated, dual dia¬ 
phragm, dc motor driv¬ 
en, 28 v, 1 w, 0.178 x 
0.178 x 0.127 m, 6.8 
kg. 

j 

j 


^Filter absorber system is a required accessory. Contact firm for details. 
>Motor required. 















540 


DETECTOR COOLING SYSTEMS 


12.2.3. Expansion-Engine Coolers. The expansion-engine cooler is based on the 
adiabatic and reversible expansion of a gas in doing work on an external load. This 
expansion results in a reduction of the gas temperature due to the decrease in the 
internal energy of the gas by the amount of the work done. A reversible expansion 
of a gas from a high to a low pressure can be produced by either piston- or turbine- 
type expanders; however, more development work has been done on the piston type 
because of its ability to handle the small gas flows required. If it is necessary that 
liquid coolant be produced by an expansion engine, the expanders are used only to 
produce the low temperature required for liquefaction, never to form the liquid. The 
formation of liquid in the expander is a highly inefficient process because the liquid 
wets the wall of the cylinder or turbine, thereby facilitating the flow of heat from the 
hot walls to the working substance and introducing considerable irreversibility into 
the cycle. In addition, the liquid introduces serious mechanical instability in the 
expander and the expander must work much harder to move the vapor-laden gas. 
The production of the actual liquid is usually carried out by a separate Joule-Thomson 
expansion valve that is supplied with gas cooled by heat exchange with a regenerative 
heat exchanger. Two types of expanders are now in use: the piston and the turbine 
types. Table 12-6 lists specific models of commercially available expansion-engine 
coolers. 

12.2.3.1. Gifford-McMahon Piston Expander. The Gifford-McMahon expander 
uses a reciprocating flow engine based on the Stirling cycle, in which the gas is passed 
back and forth through a thermal regenerator that acts as a heat reservoir. The 
Gifford-McMahon expansion engine consists of an expander-regenerator, a compressor, 
a surge tank, a supply tank, and the necessary controls (Fig. 12-11). The compressor 
is a special noncontaminating type, usually with a piston or diaphragm arrangement, 
used to maintain the coolant gas at the necessary pressure. The surge tank and 
pressure regulators provide a continuous, constant-pressure gas supply for the cooler. 
The supply tank is provided as a high pressure source of make-up gas to maintain the 
proper amount of gas in the system for long periods. 
















































TYPES OF DETECTOR COOLING SYSTEMS 


541 


Table 12-6. Expansion Engine Coolers 


Manufacturer 

Model 

Cycle 

Coolant 

Operating 
Pressure 
[(kg/m z ) x 10 3 ] 

Flow 

Rate 

Ai Research 

Ne-H 2 

Closed, Claude & 

Ne 

36.7 

0.36 kg/m 



Joule-T 

lomson 

He 

376 

0.015 kg/m 


134708-1-1 



Ne 

21.05 

0.036 kg/m 


n 2 



n 2 

49.3 

0.08 kg/m 

Air Products 

Piston 

Clo 

sed 





Turbine 






Arthur D. Little 

Cryodyne 

Closed, Gifford- 

He 




CRYR-200 

McMahon & 






Joule-Thomson 





Minicooler 

Closed 


He 

176 

31.1 liter/min at 60° 


MNR-11 

(laboratory 



19.7 liter/min at 80° 



use only) 




Cryogenerators 

Cryogenerator 

Closed, Stirling 

He/N 2 

84.5 


Div. Norelco 

42300 






Malaker Labs" 

Mark V 

Closed, S 

tirling 

He 

176 



Mark VII 






Norden 

Integrated 

Closed, Gifford-McMahon 

He 

210.5 

8.5 liter/min 


Detector 







Cooler 






Hughes 

HAC MK 11/12 

Solvay engine 

He 4 

200/60 

0.20 g/min 

Aircraft Co. 


with compressor 





HAC MK 11/30 

Solvay engine 

He 4 

200/60 

0.17 g/min 



with compressor 





HAC MK III/4 

Solvay engine 

He 4 

300/150 

2.0 g/min 



plus Joule- 



Solvay 




Thomson 



160/10 

0.2 g/min 






Joule-Thomson 



HAC MK IV/80 

Stirling refrigerator 

He 4 + air 

150/75 He 




with air-liquid 






transfer loop 


10 Air 



HAC MK IV/30 

Stirling refrigerator 

He 4 + Ne 

150/75 He 




with neon-liquid 


10 Ne 




transfer loop 






















542 


DETECTOR COOLING SYSTEMS 


Table 12-6. Expansion Engine Coolers ( Continued ) 



Operating 

Cooling 

Cooldown 


Expander 


Manufacturer 

Temperature 

(°K) 

Capacity 

(w) 

Time 

(min) 

Construction 

Size 

(m) 

Weight 

(kg) 

AiResearch 

27.16 

16 


Turbine 

0.089 x 



4.2 ±0.1 

1 




0.038 diameter 



25.66 

1 







78 

10 






Air Products 




Piston 







Turbine 



Arthur D. Little 

4.3 

0.25 


Piston 

0.33 x 0.457 

61.25 


17 

0.04 




x 0.915 



45 

0.80 







60-300 

0.05 at 60° 

5-8 



0.038 x 0.038 




0.17 at 80° 

3-5 



x 0.07 with 
0.0794 x 

0.0286 diameter 
motor 


Cryogenerators 

25-300 

1 at 30° 

5-7 

Displacer 

0.1015 x 0.305 

4.536- 

Div. Norelco 


10 at 80° 




x 0.127 

5.44 with¬ 
out motor 

Malaker Labs" 

95 

1 

12 

Displacer 

0.1015 x 0.152 








x 0.203 



70 

3 




0.356 x 








0.127 diameter 


Norden 

30-35 

0.05 

10 

Piston 

0.178 x 

0.51 







0.0381 diameter 


Hughes 

10-15 

0.4 at 12.0°K 

20 min with 

Piston 

engine 

0.05 diameter 

1.0 

Aircraft Co. 



typical CuGe 
installation 



x 0.3 



20-40 

2.0 at 30° K 

5 min with 

Piston 

engine 

0.05 diameter 

1.0 



1.0 at 25° K 

typical HgGe 



x 0.25 




0 at 20°K 

installation 






2.0-4.2 

2.0 at 4.2°K 

30 min 

Piston 

engine 

0.11 diameter 

10.0 



±5.0 at 15° K 
±12.0 at 50° K 
±60.0 at 150° K 

unloaded 



x 0.6 



70-90 

9.0 at 78° K 

<5 min with 

Piston 





(see Remarks) 

typical 

detector flask 






28 

2.0 at 28° K 

< 10 min with 

Piston 





(see Remarks) 

typical flask 

























TYPES OF DETECTOR COOLING SYSTEMS 


543 


Table 12-6. Expansion Engine Coolers ( Continued) 


Compressor 

Cooling Head 

Remarks 

Ne=single stage, oil-lubricated centrifugal, 

Ne=regenerative heat ex- 

16-w Claude by-pass expander precooler 

ac motor driven, 200 v, 400 cps, 4.9 hp, 
0.152 x 0.152 x 0.279 m. 

He=two stage, nonlubricated piston, ac 
motor driven, 200 v, 8 A, 500 cps, 1.75 hp. 

changer 

He=bundle-type counter¬ 
flow and plate-fin coun¬ 
terflow 

with 1-w Joule-Thomson cooler. 

0.61 X 0.356 m diameter. 




Single stage, nonlubricated piston, ac mo- 

Bundle-type counterflow 

Two-stage neon cooler. 

tor driven, 200 v, 400 cps, 0.42 hp, 0.406 x 


Two-stage nitrogen cooler. 

x 0.203 m diameter. 

Single stage, nonlubricated piston, ac mo¬ 
tor driven, 200 v, 400 cps, 1.0 hp, 0.61 
x 0.356 m diameter. 




Single stage, oil activated, nonmetallic dia- 

Bundle-type counterflow 

Not now used for detector cooling. 

phragm, motor driven. 

Single stage, centrifugal compressor, driven 
by expander. 




Two stage, oil-lubricated piston, ac motor 
driven, 440 v, 4 kw, 60 cps, 3 </>, plus 110 v, 


Designed for masers 
detectors. 

can be modified for 

110 w, 60 cps, 1 <t> for expander, 0.71 x 
0.762 x 1.905 m, 352.5 kg. 




MNRC-1, single stage, oil-lubricated piston, 
ac motor driven, 115 v, 350 w, 60 cps, 1 </>, 
plus 110 v, 110 w, 60 cps, 1 <t> for expander, 

Stainless steel cylinder 
0.1905 liter x 0.00788 m 
diameter. 

Laboratory use only; 

requires 10-p. filter. 

0.305 x 0.61 x 0.356 m, 54.5 kg. 




_ 

Evacuated container 

Requires 1750 rpm. motor, power input 200 


0.0191 m diameter. 

w at 30° K, 100 w at 100° K. 

- 

Insulated container 

0.0127 m diameter. 

Includes ac motor, 110 v, 1.1 A, 125 w, 
60 cps, 1 <f>. 


Insulated container 

0.0127 m diameteer. 

Includes ac motor, 208 v, 0.8 A, 140 w, 
400 cps, 3 </>. 

Single stage, nonlubricated diaphragm ac 

Integral with expander. 

Requires surge tank, makeup supply, 

motor driven, 0.305 x 0.089 m diameter, 


and pressure regulators. 

4.54 kg. 




Single stage 3-piston dry compressor 0.11 m 

0.0065 m diameter 

All performance figures given for 170°F 

diameter x 0.35 m long, 12.0 kg, 500 w 

cylinder 

(350°K), 50,000 ft altitude ambient with 

electrical input. 


self-incorporated heat rejection. Size and 
weight figures do not include heat-rejec¬ 
tion mechanism. 

Same as MK 11/12 compressor above. 

0.015 m diameter 

MK II can be modified for efficient cooling 


cylinder 

at higher temperature. 11/12 and 11/30 
will refrigerate at any temperature above 
nominal without modification. 

Solvay compressor: single stage 4 piston, 
0.3 x 0.3 x 0.8 m, 5 kw input, 25 kg. Joule- 

0.075 m diameter 
cylinder. 

Not suited for usual infrared systems. 

Thomson compressor: 2 stage 4 pistons, 
0.3 x 0.3 x 0.75 m, input 1.5 kw, 25 kg. 




Total system: 0.075 x 0.125 x 0.2 m, 450 w 

Glass dewar as needed. 

Replaces typical LN 2 transfer system; semi- 

input, 8 kg. 


closed air loop uses environment as res¬ 
ervoir with self-filtered intake. Multi¬ 
detector systems can be cooled. 

Total system: 0.075 x 0.125 x 0.2 m, 450 w 

Glass dewar as needed. 

Similar to MK IV/80 with sealed neon 

input, 8 kg. 


loop. Both MK IV systems are rated with 
3 m transfer line length. 


“Two-stage cooler, consisting of two displacer expanders on a common crankshaft, for operation between 30°K 
and 90° K under development. 














544 


DETECTOR COOLING SYSTEMS 


12.2.3.2. Displacer Piston Expander. The displacer expander (Fig. 12-12) is a 
modification of the reciprocating flow engine based on the Stirling cycle, in which the 
gas is passed back and forth through a thermal regenerator that acts as a heat reservoir. 
The displacer expansion engine is considerably simpler than a comparable Gifford- 
McMahon expansion engine in that there are no valves and a separate compressor is 
not required. The compression of the gas is produced by the motion of the main piston 
and the displacer piston in coming together. The compressed gas flows through the 
water jacket, where the heat of compression is removed, to the regenerator. The ex¬ 
pansion of the gas takes place against the displacer piston in the space at the top of 
the cylinder. The main drawbacks of the piston-type expander are the wear of the 
sliding surfaces and the valve leakage when small volumes of gas are used. In addition, 
the reciprocating-flow engines are limited by the heat capacity of the regenerator 
material. Lead is the most common material used, since it can store significant 
amounts of heat down to approximately 14° K; other materials such as aluminum, zinc, 
brass, and bronze have no meaningful heat capacity below 35° K. 

12.2.3.3. Turbine Expanders. Turbine expanders have not been used much because 
they require relatively larger flows of gas than piston compressors and expanders. 
Infrared detector coolers requiring low refrigeration loads, and operating at the normal 
atmospheric boiling point of the gas, usually have flow rates much too low for turbine 
machinery. Turbine machinery is more applicable to systems using depressed boiling 
points, where overall life and reliability are important considerations. Progress has 
been made recently in the development of turbine expanders that operate at low pres¬ 
sure ratios and flow rates, and that are capable of providing sufficient refrigeration 
to liquefy cryogenic gases. A schematic diagram of a proposed helium refrigerator 
using a turbine expander for cooling a maser to 4.2° K is shown in Fig. 12-13. 

12.2.4. Thermoelectric Coolers [1, 8-12]. Table 12-7 lists specific models of com¬ 
mercially available thermoelectric coolers. 

The basic principle of the thermoelectric or Peltier cooler is the Peltier cooling effect, 
which is caused by the absorption or generation of heat when a current passes through a 
junction of two dissimilar materials. The rate of pumping of heat is directly propor¬ 
tional to the current, and the constant of proportionality is known as the Peltier coeffi¬ 
cient, 7 r. It represents a potential difference, which is determined by the Fermi energy 
plus a transport energy. Thus, for a current /, the rate of pumping of heat by the Peltier 
effect, Q n , is 

Qn = TTl ( 12 - 1 ) 

The Seebeck coefficient of a couple, S, is the ratio of the thermal emf to the tempera¬ 
ture difference which produces that emf. For a single material, a quantity s, called 
the Seebeck coefficient of the material, can be defined in such a way that the Seebeck 
coefficient of any couple is given by the difference of the Seebeck coefficients of the 
two materials from which it is constructed: 

S = s 2 — s i 

The Seebeck coefficient of a p-type material, s p , and the Seebeck coefficient of an rc-type 
material, s„, must have opposite signs. Conventionally, s p is taken as positive and 
s„ as negative. Thus, for a couple composed of a p-type and an n-type material, the 
difference s p — s„ is actually the sum of two positive numbers. In order to avoid diffi¬ 
culties with signs it is often convenient to use absolute values: 


\s\ = |s„| + M 


TYPES OF DETECTOR COOLING SYSTEMS 


545 


Expansion Space 
Displacer Piston 
Regenerator 
Cooler 


Evacuated Housing 

Detector Element 


Cooling Water Out 

Main Piston r- 1 



f 

\ 

L 

V 

J 

/ 

i 


Fig. 12-12. Displacer expansion engine. 


COMPRESSOR 



MAJOR INPUT PARAMETERS 

r c = Compressor pressure ratio 

Tl^ = Compressor overall efficiency 

T| = Aftercooler exit temperature 

P8 = Evaporator exit pressure 

Qi = Primary refrigeration load 

Qz,Qa = Secondary cooling loads 

ei,€3,€s = Effectiveness of first, third 
and fifth heat exchangers 

AT, = T, - T, 

AT 2 = T 4 - T, | 

\lt, = Alternator efficiencies 


Fig. 12-13. Analytical model for two expander Claude Cycle Analysis Computer Program. 






















































































































546 


DETECTOR COOLING SYSTEMS 


Table 12-7. Thermoelectric Coolers 


Manufacturer 

Model 

Construction 

Nominal 

Load 

Cooling 

(mw) 

Cold 

Junction 

Temperature 

(°K) 

Heat-Sink 

Load 

(w) 

Hot 

Junction 

Temperature 

<°K) 

Size 

(m) 

Energy 

D5-60 

One-stage 

2.6 

240 


300 

0.044 x 0.025 x 0.0094 

Conversion 









D9-60 

One-stage 

5.3 

240 


300 

0.044 x 0.025 x 0.00686 


E9-60 

One-stage 

11.2 

240 


300 

0.053 x 0.0348 x 0.00686 


D-series, 

Two-stage 

d 

207-225 

5-12 

300 

0.044 x 0.025“ 


E-series 






0.053 x 0.0348-- 


D-series, 

Three-stage 

d 

200-205 

12 

300 

0.044 x 0.025“ 


E-series 






0.053 x 0.0348“ 

Materials 


Hot & cold stage 

4 

233 


293 

0.044 x 0.119 o.d. 

Electronics 


with 15 two-stage 






(Melcor) 


modules 






Pesco 

094429 

Four-stage cascade 

15 

195 

12 

300“ 

0.07625 x 0.117 x 0.152 


094438 

Four-stage cascade 

15 

195 

12 

300“ 

0.043 x 0.061 x 0.035 


094446 

One-stage 

50 

242 in vacuum 

3 

300“ 

0.01745 + 0.0254H 





252 in N 2 








259 in air 





094447 

Two-stage cascade 

20 

223/251 

4/4 

300/351“ 

0.0413 x 0.0524 x 0.0381 


094492 

One-stage, 

200 

252 

3 

300“ 

0.0127 x 0.0088 x 0.0127 



eight couples 







094493 

Three-stage cascade 

15 

208 

1 

300“ 

0.0305 x 0.0508 x 0.0302 


094567 

One-stage 

10 

243 

0.5 







253 


300" 

0.0159H 





258 





094568 

One-stage, 

20 

243 

2.8 





three couples 


253 


300“ 

0.0206H 





258 





094575 

One-stage 

30 

253 

1 




094618 

Two-stage 


203 

1.9 

300“ 

0.0381H 







300“ 

0.0305 x 0.0508 x 0.0478 

Radiation 

Thermo- 

Three-stage 


195 



0.0401H 

Electronics 

electric 








cooler 







Texas 

Thermo- 

One-stage, 

80 

258 

1 

300 

<0.00254H 

Instruments 

electric 

two couples 







cooler 







Westinghouse 

WX-814 

One-stage 

10 max 

b 

36 max 

373 

0.0396 x 0.0396 x 0.0127 




5 nom 




lugs & leads 


WX-816 

One-stage 

10 max 

b 

36 max 

373 

0.0396 x 0.0396 x 0.0454 




5 nom 




lugs & leads 


WX-817 

One-stage 

1 

AT = 55 



0.01575 x 0.0159 x 0.0127 








lugs & leads 


WX-824 

Similar to WX-814 




523 



WX-825 

Similar to WX-814 




623 



WX-826 

Similar to WX-816 




523 



WX-827 

Similar to WX-816 




623 



“Hot-junction temperature given for rating purposes only; all units capable 
of operating with hot junction temperature of 373°K. 


- rr esungnuu.se 

WX 814-E 
WX 814-F 


WX 814-J 


use Model 

AT 

WX 

816-E 

35° C 

WX 

816-F 

40° C 

WX 

814-G 

45° C 

WX 

814-H 

50° C 

WX 

814-J 

55° C 

















TYPES OF DETECTOR COOLING SYSTEMS 


547 


Table 12-7. Thermoelectric Coolers ( Continued) 


Diameter 

(m) 

Mounting Area 

Vacuum 
Pressure 
(mm Hg) 

Voltage 

(volt) 

Current 

(amp) 

Ripple 

Maximum 

(%) 

Weight 

(kg) 

Remarks 

— 

0.0206 x 0.025 

Depends on 

1.2 

5 

10 

0.017 

Electrically isolated heat transfer 



application 





surfaces. 


- 

0.0206 x 0.025 



1.2 

9 

10 

0.014 



- 

0.03 x 0.0348 



2.5 

9 

10 

0.025 



- 

c 



c 

c 

10 

C 

Long time constant, others available 


c 







that operate into a low-temperature 


c 







heat sink. 



c 



c 

c 

10 

c 



0.0525 

15 coplanar 

Depends on 

28 

1.8 

10 

0.453 

Used to cool vidicon tube; consists of 


heat pads 

application 





15 two-stage modules, 8 modules in 









cold stage, and 20 modules in hot 









stage. 


— 

0.0161 m 2 

io - 6 

0.6 

20 

10 

25 

Includes integral detector. 

0.0356 

0.0161 m 2 

10« 

0.6 

20 

10 

2.5 



0.025-0.0253 

0.0111 m diameter 

Depends on 

0.4 

7 

10 

0.5 





application 







0.0325 

0.0322 m 2 

io- 3 -io-« 

1.2 

3 

10 

2.0 

Used for cooling of five diodes. 

— 

0.0322 m 2 

Depends on 

0.85 

3.5 

10 

0.2 





application 







0.0239 

0.008 m 2 

10 -• 

0.17 

6 

10 

1.0 



0.00396 

0.00396 m diameter 

Depends on 

0.1 

5 

10 

0.1 





application 







0.00762 

0.00762 m diameter 

Depends on 

0.4 

7 

10 

0.2 





application 







0.00762 

0.00508 m diameter 

Air 


0.2 

5 

10 

0.5 

Fast time constant, cooldown 5 sec. 

0.0239 

0.00318 x 0.00635 m 

io-« 

0.19 

10 

10 

1.0 



0.0302 



2.0 

2.5 



Includes integral detector. 

0.0028 




0.2 

<4 

<15 


Horizontal cold 

mounting surface 









reaches 63% of final AT after 1.5 min 









with no load. 


_ 

0.148 m 2 

Air** 


1.2 

20 

10 

2.8 

Horizontal cold 

mounting surface 









reaches 63% of final AT after 1.5 min 









with no load. 


_ 

0.103 m 2 

Air" 


1.2 

20 

10 

3.1 

Vertical cold mounting surface 









reaches 63% of final AT after 1.5 min 









with no load 



0.043 m 2 

Air d 


1.25 

6 

10 

1.5 




c Depends on AT and cooling load. d Coolers rated with cold junction and device being cooled surrounded 

by insulation at normal ambient temperature (T A ) of 35°C. 




















548 


DETECTOR COOLING SYSTEMS 


The Seebeck and Peltier coefficients are related by the first of the Thomson rela¬ 
tions [13 or 14]: 

77 — ST ( 12 - 2 ) 

It follows that 

Qn = ST I ( 12 - 3 ) 

where T is the temperature of the cold junction when the couple is used for cooling. 

Evidently, then, Peltier cooling will be more effective with materials which exhibit 
large Seebeck coefficients. It is evident also that cooling will be more effective when the 
joule heating, I 2 R, is a minimum. The current, /, cannot be made small without re¬ 
ducing the rate of heat pumping (see Eq. 12-1). Therefore it is necessary to keep the 
resistance low, which can be accomplished by the use of material with low electrical 
resistivity, p. A third important property is the thermal conductivity k. Clearly 
this parameter should be kept as small as possible, since it would be of little value to 
pump heat from one region to another if most of it could flow back again. 

Therefore, the three following parameters serve to characterize a material for its 
cooling capabilities: 

s = Seebeck coefficient (v °K _1 ) 

p = electrical resistivity (ohm-cm) 

k = thermal conductivity (w cm 1 °K _1 ) 

These three quantities vary with temperature, and for accurate calculations the 
variations must be taken into account. Furthermore, variations of s with T give rise 
to an additional thermoelectric effect, the Thomson effect, and a rigorous treatment 
must take into account the Thomson heat which arises when a current and a parallel 
temperature gradient exist in a material for which s varies with T. However, for 
the purposes of this discussion of Peltier cooling, s, p, and k will be assumed constant 
over the temperature range under consideration, and hence the Thomson effect will 
not affect the problem. 

12.2.4.1. Figure of Merit. The three material properties s, p , and k frequently appear 
in the same combination in discussions of thermoelectronic devices. This combination 
is usually referred to as the figure of merit of the material and is denoted by Z, where 


Z = — (°K- 1 ) 

px 


(12-4) 


In general, the larger the figure of merit, the more useful the material for thermo¬ 
electric applications. At present, the best values of Z for semiconducting materials 
are slightly greater than 3 x 10 -3 /°K. (The best values for metals are about 0.1 
X 10 _3 /°K.) A figure of merit of 3 x 10 _3 /°K makes it possible, for instance, to pump 
heat from ice to steam with a single-stage device. 

The figure of merit of a couple is defined by 

Z ' = M < 12 - 5) 

where the material constants p and k of Eq. (12-4) have been replaced by the elec¬ 
trical resistance R and the thermal conductance K. 

The maximum possible value of Z c for any couple composed of two given materials 
([14], page 11) is given by 


Z max 


vVpTkT + Vp 2 K 2 / 


( 12 - 6 ) 






TYPES OF DETECTOR COOLING SYSTEMS 


549 


It should be noted that Z mnjr depends only on the properties of the materials, whereas 
Z c depends also on the dimensions of the couple. Z c approaches Z max only under ideal 
conditions when the relative values of the various dimensions have been properly 
adjusted. 

If the values of p and k are the same for the two arms of the couple, it can easily 
be seen that Z c = Z max when the dimensions of the two arms are equal. If, in addition, 
the couple is made from p-type and rc-type materials such that s p — —s» = s, then Z c = 

Z max S 2 / p K. 

12.2.4.2. Performance of a Peltier Couple. This treatment is not intended to be an 
exact physical or mathematical description, but only an outline of the important fea¬ 
tures of Peltier couples. The numerical examples given are evaluated from approxi¬ 
mate expressions because the errors are small and the properties of the thermoelectric 
materials are not known with sufficient accuracy to warrant exact calculations. 

It will be assumed that the couple has Seebeck coefficients s„ and s ,,, electrical re¬ 
sistivities p„ and p p , thermal conductivities k„ and k p , total series electrical resistance 
R, and total parallel thermal conductance K between the ends, which are at tempera¬ 
tures T c and TV The temperature difference Th — T c will be called AT. Figure 12-14 
is a sketch of the idealized Peltier couple that will be discussed. The conductor con¬ 
necting the thermoelectric materials is assumed to have zero Seebeck coefficient, zero 
electrical resistivity, and infinite thermal conductivity. The thermal and electrical 
resistances at these connections are assumed to be zero. When it is also assumed 
that the properties of the material are independent of temperature, it is possible to 
give a complete description of the equilibrium performance of the couple. It will be 
shown that s„, s p , R, K, and T c completely define the maximum temperature difference 
attainable with this particular couple. However, this is not necessarily the maximum 
temperature difference attainable with the materials from which the couple is con¬ 
structed. 



Assume 

s = p = 0 
max max 

K =00 

max 


Fig. 12-14. Peltier couple with notation and definitions. 


Four quantities are of interest: the pumping current I, the heat-pumping rate Q, 
the temperature difference AT, and the coefficient of performance £ (to be defined 
later). Only two of these are independent. However, generally two conditions are 
of greatest interest, namely, the pumping of the maximum amount of heat, and the 
most efficient pumping of heat. Each of these conditions fixes the magnitude of the 
current so that, for a given couple, the heat-pumping capacity and coefficient of per¬ 
formance can be represented as functions of AT and T c only. 


























































550 


DETECTOR COOLING SYSTEMS 


12.2.4.3. Heat Balance at the Cold Junction. In the couple shown in Fig. 12-14, 
the current passing from ntoMtop will remove heat from M. The rate of heat removal 
by the Peltier effect is 0* = ( s p T c I — s n T c I) = ST C I. 

It can also be shown that the uniform generation of joule heat throughout the thermo¬ 
electric material results in a flow of heat to each of the junctions at the rate 0j = A I 2 R. 
(It is assumed that there is no heat transfer through the sides of the thermoelectric 
arms.) Finally, the heat conducted back to the cold junction from the hot junction is 
0/ = #AT. 

At equilibrium, the difference between the flow of Peltier heat, 0*> from the cold 
junction and the flow of joule heat, 0j> and conducted heat, 0/> to the cold junction 
represents the rate 0 at which the couple pumps heat. Thus, 

0 = 3*- (4j + Qf) = ST C I - \PR — K AT ( 12 - 7 ) 

Because of the power dissipated in the thermocouple (P) during the heat-pumping 
process, 0 + P watts must be rejected at the hot junction for 0 watts absorbed at the cold 
junction. 

12.2.4.4. Maximum Rate of Heat Pumping. The current which produces the maxi¬ 
mum rate of heat pumping is easily obtained maximizing Q (by setting dQ/dI=0). The 
result is that the current for maximum steady-state heat pumping, /<>, is 

Id = ST JR (12-8) 

which is proportional to the temperature of the cold junction and independent of the 
load or temperature difference. The maximum rate of heat pumping is obtained by 
substituting Eq. (12-8) into Eq. (12-7). The result is 

i .Q 27 1 2 

Clma,=-— 5 ^-K\T (12-9) 

A K 


If the cold junction is insulated so that no heat is absorbed from its surroundings, 
AT will rise to the maximum value that this couple can provide, AT max . Thus, 


AT 


max 


1 S 2 TJ 

2 RK 


( 12 - 10 ) 


The figure of merit of the couple, Z c , may then be defined as 

Z c = SJRK (12-11) 

The maximum temperature difference that this couple can maintain is therefore 

A Tmax = \Z C TJ (12-12) 

With optimum design, a couple made of the same materials can maintain a tempera¬ 
ture difference A T M , given by 

AT m = \ZmaxTj (12-13) 

where Z max is defined by Eq. (12-6). In general, A T max is slightly less than A T M . 
Equations (12-9) and (12-10) show that the rate of heat pumping may be written 

Q max — K{ATmax AT) = K AT max f 1 —- ^ (12-14) 

\ Ai max' 

which decreases linearly from K A T ma x at AT = 0 to 0 at AT = A T max . 





TYPES OF DETECTOR COOLING SYSTEMS 


551 


The coefficient of performance, £, is defined as 




(12-15) 


where P is the power required to pump heat at the rate 0 , and is given by 

P = IS AT + PR 


(12-16) 


= 2 K\T 


max 



(12-16') 


By combining Eq. (12-14), (12-15), and (12-16'), one finds the coefficient of performance 
for maximum heat pumping to be 



(12-17) 


This has a maximum value of 1/2 and decreases to zero when AT = A T max . 
Summarizing; it is now possible to obtain: 

(а) The maximum no-load temperature difference, AT max , from Eq. (12-12). 

(б) The current required for maximum cooling, 7$, from Eq. (12-8). 

(c) The amount of heat pumped, 0 wax , for a given AT, from Eq. (12-14). 

( d) The coefficient of performance, £<j, for a given AT, from Eq. (12-17). 

"Typical values of Qmax, £<}, and 7<j, for material constants somewhat less than 

the best attainable are presented in Fig. 12-15. 


(0 


a 

e 


? © 'o 


,<y •<? 












J Q 





r • 




• ^ ' 

's 





Q ' 
m 












AT 0 

10 20 

30 

40 

50 

60 

T C (°K) 300 

290 280 

270 

260 250 

240 

T c (°C) 27 

17 7 

-3 

-13 - 

23 

-33 

Fig. 12-15. 

Maximum rate of heat 

pump- 


ing, current required for maximum heat¬ 
pumping capacity, and coefficient of per¬ 
formance for maximum heat pumping as a 
function of T c (or AT) for T/, = 300° K [8]. 


12.2.4.5. Maximum Efficiency. The expression for the current required for maxi¬ 
mum efficiency may be obtained by maximizing the coefficient of performance with 
respect to current, using Eq. (12-7), (12-15), and (12-16). The result obtained shows 
that the current giving maximum coefficient of performance, h, is 

AT 

h = T^— A (12-18) 

max 


where A is a factor close to unity given by the relation 


Tc + VTc 2 + 2 Tc AT max + AT ATmax _ „ , T ma x ~ AT 

A = ---= 1 H-—- 

2 T c 


2 T c + AT 


(12-19) 























552 

Therefore, 


DETECTOR COOLING SYSTEMS 


h=h 


AT 


AT 


( 12 - 20 ) 


max 


Note that for constant T c the current for maximum heat pumping, /<}, is constant and 
the current for maximum efficiency, /$, varies almost linearly with AT. 

From Eq. (12-7) and (12-8), the rate of heat pumping is found to be 


Qt = KAT 


(2 A - 


AT 


A 2 - 1 


AT max 

where A is defined by Eq. (12-19) and is nearly unity, so that 

AT 


( 12 - 21 ) 


Qi = KAT 


/ _ AT \ 
\ AT mar-) 


( 12 - 22 ) 


This may be written in terms of the maximum heat pumping rate as 

AT 

Qi = Qmax — - (12-23) 

A 1 max 

and will be zero for AT = 0 and AT= AT,„ flJ , with a maximum at AT= AT maJ /2, where 

Q$ = Q max/ 2. 

The maximum coefficient of performance, from Eq. (12-21), (12-18), (12-16), and 
(12-15) becomes 


_ 1 / 1 A AT \ /AT AAT \-» 

UaX 2 V A ATmax/ \T f ATmax) 


and again, since A is nearly unity, this simplifies to 


C 


max 


1 / AT \ / AT AT y 1 

2 V ATmax) \ATc ATmax) 


(12-24) 


(12-25) 


This may be written 


£ 


max 


ATmax 1 

AT ’ 2 V 


AT \ / _ AT\ / AT mas — AT \ -1 
ATmax) \ Th) \ T/, / 


(12-26) 


which, from Eq. (12-17), is very nearly 

„ ATmax „ 

£max = ' Cq (12-27) 

Note that two approximations are used in arriving at this simple form. The exact 
expression given by Eq. (12-24) can be approximated by Eq. (12-25), since A is very 
nearly unity. Equation (12-26) is merely Eq. (12-25) written in a different form. 
Finally, Eq. (12-26) is written as Eq. (12-27) by assuming that 

ATmax ~ AT << Tn 

The ratio AT/ A T ma x is seen to be very nearly the ratio of the coefficients of perfor¬ 
mance for maximum cooling rate and maximum efficiency. Figure 12-16 shows the 
values of Cmax, h, and Qj when the couple used for Fig. 12-15 is used under conditions 
of maximum coefficient of performance. The curves were obtained using Eq. (12-20), 
(12-23), and (12-27). For very small AT, Eq. (12-26) was used instead of Eq. (12-27). 

















TYPES OF DETECTOR COOLING SYSTEMS 


553 



Fig. 12-16. Maximum coefficient, of per¬ 
formance, current required for maximum 
coefficient of performance, and heat-pumping 
rate for maximum coefficient of performance 
for typical couple [8]. 77, = 300° K. 


Analytical expressions for Q and £ are not difficult to obtain in simple form for other 
values of current. Some idea of the way in which Q and £ vary with current and 
temperature difference may be obtained from Fig. 12-17, where Q and £ are plotted 
against I with AT as a parameter. Again, the numbers refer to the couples used 
for Fig. 12-15 and 12-16. 



Fig. 12-17. Rate of heat pumping 
and coefficient of performance as 
functions of current for various tem¬ 
perature differences with Th = 300° K 
[ 8 ]. 























































554 


DETECTOR COOLING SYSTEMS 


12.2.4.6. Single-Stage Coolers [8]. A single-stage thermoelectric cooler consists 
of a p-type and an rc-type semiconductor connected together by a metallic conductor 
(Fig. 12-18). An external battery causes the flow of a current; flow produces a tem¬ 
perature difference between the two junctions by absorbing heat at one and releasing 
it at the other. The performance of a thermoelectric couple as a function of temperature 
difference is shown in Fig. 12-19. The hot junction is held fixed, and the temperature 
is varied by changing it at the cold junction. For each change in temperature, the 
applied voltage has been adjusted to maximize the coefficient of performance. The 
two main operating modes for any thermoelectric cooler occur at the points where 
both the heat-pumping rate and the coefficient of performance are maximum. 


.Cold Junction 




Fig. 12-19. Performance of thermoelectric couple as 
a function of temperature difference [9]. 



















TYPES OF DETECTOR COOLING SYSTEMS 


555 


12.2.4.7. Single-Stage, Multicouple Coolers. Thermoelectric couples of similar 
material can be arranged in thermal parallel to increase their heat-pumping capacity. 
When identical couples are placed in thermal parallel (Fig. 12-20) and supplied with 
equal currents, they pump as many times more heat at the same temperature difference 
and with the same coefficient of performance as there are couples. To ensure that 
the couples are supplied with equal currents, they are connected in series electrically. 
Thin foils or films, or materials of good thermal conductivity, are used as insulation 
between couples since they do not increase the thermal resistance significantly. 


Mounting Surface 



12.2.4.8. Multistage Coolers. Thermoelectric couples are connected in thermal 
series or cascaded for two purposes: to provide a temperature difference greater than 
that obtainable from a single couple, or to achieve a higher coefficient of performance 
for a given heat load at an established temperature difference. The limit on the coeffi¬ 
cient of performance is related to the limit on the maximum temperature difference by 



(12-28) 


C max — 


A T 


max 


AT 


- (l- 

2 \ A T max 


1 - 


AT 

T„ 


1 + 


A T max - AT 
T h 


(12-29) 


The coefficient of performance for cascaded couples is given by 


t = 





(12-30) 


The results obtained from the use of a two-stage cooler may be illustrated by the 
the following example. A single-stage cooler made from materials such that Z c = 
2 x 10 _3 /°K (Sec. 12.2.4.1) and operating between 260°K and 300°K is replaced by 
two stages, each operating with AT = 20°. When the currents are adjusted for maxi¬ 
mum efficiency, the coefficients of performance are 0.30 for the single-stage cooler 
and 0.42 for the two-stage cooler. However, if the currents are adjusted for maxi¬ 
mum heat pumping, the coefficients are 0.18 for the single-stage cooler and only 0.06 
for the two-stage cooler. 

Further improvement could be obtained by adding a third stage, but for the above 
example the improvement is small (about 10%). In general, the improvement for 





















556 


DETECTOR COOLING SYSTEMS 


two stages is greatest when AT/AT ma x is nearly unity. When ATI AT max > 1 , a multi¬ 
stage device is the only way to remove heat from the cold junction under steady-state 
operation. 

The preceding discussion has referred primarily to cascaded systems in which the 
objective is to improve efficiency with a given temperature difference. If the principal 
objective is to obtain a large temperature difference between the cooled region and the 
heat sink, each stage can be operated, inefficiently, at nearly its maximum tempera¬ 
ture difference, thereby giving AT max = ATmaXi + AT max 2 + AT max 3 + . . . + AT,„ax n 
for n stages. Since operation at with AT = AT max is inefficient, successive stages at 
higher temperatures must be designed to pump rapidly increasing amounts of heat. 
It should be borne in mind that AT,„ax is different for each of the successive stages 
even if constant material properties are assumed, because of the dependence of AT ma x 
on the square of the absolute temperature of the cold junction. 

In considering the design of multistage coolers, it is clear that it would be undesirable 
to supply power to each stage separately through copper leads from a power supply 
at a temperature of approximately 300° K. Both the thermal losses through the leads 
and the thermal insulation between stages, introduced by the required electrical in¬ 
sulation, would constitute sources of inefficiency. This inefficiency can be avoided by 
designing a network of unequal thermoelectric elements that would allow the op¬ 
timum currents to be supplied to each aim of each stage from the adjacent stages 
(Fig. 12-21). 

Figure 12-22 shows the variation of the cold-junction temperature as the input 
current is varied for a typical four-stage cascade cooler. The cold-junction tempera¬ 
ture versus cold-junction load is shown in Fig. 12-23, and the cold-junction temperature 
versus time is shown in Fig. 12-24. Because the cold-junction temperature is ex¬ 
tremely sensitive to convection loading, the thermoelectric cooler is usually mounted 
in a chamber evacuated to a very low pressure. Figure 12-25 shows the effect of the 
vacuum pressure on the cold-junction temperature. 

For the most effective cooling, the difference in the Seebeck coefficients should be as 
great as possible, the electrical resistance low, and the thermal conductivity small. 
In addition, consideration must be given to the optimum cross sections of the arms of 



Fig. 12-21. Ten couple, three- 
stage cascade cooler. 



INPUT CURRENT TO COOLER 
(dc amp) 


Fig. 12-22. Cold junction tem¬ 
perature versus input current 
with hot junction at +27° C and 
no thermal load [11]. ( Copy¬ 

right Pesco Products.) 
































TYPES OF DETECTOR COOLING SYSTEMS 


557 



0 10 20 3 0 40 50 60 70 

COLD JUNCTION LOAD (raw) 


Fig. 12-23. Cold junction tem¬ 
perature versus cold junction load 
with hot junction at +27°C [11]. 
0 Copyright Pesco Products.) 


a 

o__ 

w 

K 

O 

H 

< 

K 

W 

a, 

S 

w 

H 

i 



VACUUM PRESSURE (mm of Hg) 


Fig. 12-25. Cold junction tem¬ 
perature versus vacuum pressure 
with hot junction at +27° C and 
no other thermal load [11]. 
0 Copyright Pesco Products.) 



Fig. 12-24. Cold junction tem¬ 
perature versus time after opti¬ 
mum current is applied [11]. 
(■Copyright Pesco Products.) 


Infrared Transparent Window 
j ^^^Detector Element 
>///>/> /// [ / 1' /Sr? //; • / • . 




<; 


zzzpz. 

sis 




U 


:i 


- 




- Glass Wall 
Conducting Strip 

Coolant Well 


Platinum Lead 


Kovar Lead 


Fig. 12-26. Construction of a 
single-cell detector dewar. 


the couples, heat exchange at the hot and cold junctions, electrical and thermal re¬ 
sistance of the junctions and conductors, power-supply problems, and changes in the 
heat load or supply current. The design of thermoelectric coolers is given in [8] and 
[ 121 . 

12.2.5. Dewars. Most detectors that require cooling are mounted in a double-walled 
vacuum enclosure called a dewar. Dewars can be conveniently classified according to 
the number of coolant cells: single cell (Fig. 12-26) or double cell (Fig. 12-27). The 
dewar maintains the coolant in thermal contact with the detector, insulates the coolant 
from the environment to prevent rapid evaporation, prevents frost from collecting over 
the external surfaces of the dewar (in particular the window), and protects the delicate 
detector. Also, the vacuum construction protects the detector from the deteriorating 
effects of the atmosphere. 

12.2.5.1. Dewar Construction. A variety of detector configurations and envelope con¬ 
structions are available depending on the mode and temperature of operation. Dewars 
are usually custom designed for a particular application; however, certain stock dewars 














































































558 


DETECTOR COOLING SYSTEMS 


are available (Table 12-8). Dewars are also made by the manufacturers of detectors for 
use with their detector cells, although some companies will sell them separately. Cur¬ 
rently, dewars are constructed mainly of Pyrex glass, although metal is sometimes used. 
Pyrex glass has an advantage in that it is readily available in many sizes, but for 
dewars are constructed mainly of Pyrex glass, although metal is sometimes used. 
Pyrex glass has an advantage in that it is readily available in many sizes, but for 
helium systems it has one drawback; it is permeable to helium gas at room temperature. 
Thus, after several uses, the helium dewar must be repumped to eliminate the gas in 
the space between the walls. This difficulty can be avoided if care is taken never to 
fill the helium dewar with gas until it is at the temperature of liquid nitrogen. The 
surfaces of the dewars are normally coated with a thin layer of a substance having 
good reflection characteristics, except for a thin window or slit on opposite sides of 
the dewar walls for observation of the liquid level. In the end-looking detector, an 
infrared-transparent window is joined to the end of the dewar opposite the detector. 
In the side-looking detector, the window is incorporated in the outside wall of the 
dewar opposite the detector. Once the window is attached, the detector surface is 
no longer available for testing. A demountable dewar (Fig. 12-28) that is evacuated 
before each test is used if it is necessary that the detector surface be readily available. 
When the detector cannot be physically located in the dewar, a thermal conductor 
such as a synthetic ruby or sapphire rod provides thermal contact between the detector 
and the coolant in the dewar to assure efficient cooling. 



Infrared Transparent Window 



Fig. 12-27. Construction of a double-cell Fig. 12-28. Construction of a 

detector dewar. demountable dewar. 




























































12 - 8 . Commercially Availablf Dewars. 


TYPES OF DETECTOR COOLING SYSTEMS 


$ 

•8 

e 

* M 

St 


X> 


"S 


0 ) 

1 ~ 


£ £ 

.2 £ 
2 


CO 

o | 

45 S 

be ^ 

E TJ 

‘ c 


eg 


to 

* 

q c .§ 
d, to 2 ^ c 
Q, 3 Q 
3 * 

co 


I* -3 


I-a 
£ 

n, s 


3 

u 

t 

o> 

> 


* 

o 

T3 

3 

'£ 

JS 


E 

* 

O 

•3 

bo- 

c 

o 


B 

O 


I* 


O 


Sag 

■5 CO — 

Q 


I 

o 


£ 


2 x 3 

ID 


CO 

o 

o 

1 ■ 
IN 

O 

© 


a 

a 


2? 

a * 

k N 

ft. ■ 1* 

B, CO 


1? 

£ 


3 

1*, 


I 


^ S 

41 d 

H 


k 

<V 


£ g e 

5 .a 


■8 

— 

cc 

3 

o 


k 

$ 

45 © 

s g 

a w 


£ 


o fc E 

k. ir* 

I 


11 


bo 

Jd 


bo 

Jd 


.« 45 

c 3 

b| 

s" > 8 ■■S 

E > ® M 

*« '2 « ec 

n a j ^ 

•'* 05 £ 

g £ 

© 03 


i 
2 & 


s~ 

05 

a) 

e 

.2 


CO 

o 

d 


c 

S 

u 

0) 

13 

i s 

”3 & 

e r 

g:S 

CM ,-h 

rH 

9 2 

o 


00 

co 

ID 

00 to 

rH 

10 0 

ID 

rH G5 

CO 

CM 

05 

00 ^ 




ID 

t> 

l> CM 

to 

^ 00 

CM 

CM tO 

CM 

05 O 

CM 

CM CO 

l> 

CM 

t> 

to 00 

CM 


to 

ID CM 

00 

00 CM 

Tf 

to 


ID 

d 

d d 

d 

d d 

to 

to to 


to 

rH 

to to 


CM ID 

O 

00 00 

rH 

CM 00 

rH 

0 0 

rH 

rH O 

d 

d d 

d 

d d 


a 



a 

05 «e 

-0 

■0 -c 

t> 

CM 

co 

CM CM 

CM 

0 to 

0 

ID tO 

rH 

rH rH 

CM 

rH rH 

d 

d d 

d 

d d 


0 


0 

CD 

't ^ 

rH 

00 rH 

to 

co 06 

CM 

CD 00 


U 

fli 

£ <D 

E 1 

.2 ~ 

E « 

CO 

E 

0) 


00 

o _ 

lO J3 

§ * 


SO 

rr 


w 


in 

a> 


t> 

s? 

o 


ID 

10 

00 

o 


CM 

»D 


I 

CO 


u 

a 

13 

6 

CO 

6 


cm £ 

CM 0 ) 
CM Jl 


10 

Tf 

id 


00 


CO 


t> 

00 


CO 

CO 


<N 

IO 


CM 


U 

<v 

-h> 

0) 

E 

.2 

-3 

6 
LO £ 
CO 0) 
CO X 

9 t 

o 


.D 

jfl 

3 

> 

co 


o 

CM 

cm 


o 


CM 

CO 


CM 

CO 

t- 

o 

o 


10 


o 

o 


co 

1 

CM 


cn 

2 

o 


e 45 
§•1 


3 

&■ 


£ 

Q 


bo 

B 




13 


£ 

HH 

8 

CO 

c 

8 . 

Q. 


H 

5 

0 

0 

O 

6 

CM 



s ^ 

2 bo 

CM 

CO 

rH 




in 

t> N O 

o ^ rH 


ID © 10 CM © 
CM CM rH rH rH 


O c-l CD N C) O m(D 

KZSZSSZS 


B j«j 
§ 8 
E 2 


m n 

05 CO 

m Tf 

i-H IN 


O 

05 


Tf CO 

C£) iH 

O Tf 
(N 


60 O Q Q 


o m 

r-i o 


M 0 ) 

Z K 


-M CO 

.£ o 
Q w 


co 

o 

IN 


O 

co 


bo 

a 

t, o J£ 

3 2 - 

a c -5 

O O X, 

o o co 


co 

CD 

o 

CM 


lO 

o 


05 

05 

05 


CO 

0 ) 

# & 

2 

co 

h 

£ 

CO 

J 

C 

CO 

1 

x 

>s 

J= 

T3 

05 

c 


u 

2 


559 











560 


DETECTOR COOLING SYSTEMS 


Dewars can be used interchangeably with different coolants, but when liquid helium 
is used, certain precautions must be followed. The liquid must be protected from the 
atmosphere at all times, since at the liquid-helium temperature all other substances 
are solids. Therefore a high-capacity vacuum pump is required. Because of the low 
latent heat of liquid helium, all heat leaks into the bath must be reduced to minimum. 
This entails the use of a double-cell dewar whose outer container or shield contains 
liquid nitrogen so that the liquid helium is surrounded by a body at a temperature well 
below ambient room temperature. The dewar must be cooled down to the liquid- 
nitrogen temperature before liquid helium is placed in the inner cell to prevent the 
helium gas from permeating into the interspace and to remove the sensible heat of 
the dewar with economical liquid nitrogen rather than expensive liquid helium. Be¬ 
cause of the low latent heat of helium, a very large quantity would be required to cool 
the dewar from room temperature to liquid-helium temperature. Whereas in a regular 
dewar the interspace is evacuated to a low pressure, usually 10~ 6 mm Hg, in a helium 
dewar a considerable quantity of air, approximately 5 mm Hg, is left in this space. The 
air acts as a heat conductor and is useful in cooling the well to the liquid-nitrogen 
temperature, thus reducing the heat load on the liquid helium. When the liquid helium 
is added to the well, the air in the interspace freezes and provides an excellent insulat¬ 
ing vacuum of about 10 -10 mm Hg. 

Temperatures below the boiling point of liquid helium can be produced by controlling 
the vapor pressure through pumping. A peculiar creep property of liquid helium below 
2.19°K permits it to flow against gravity to regions that are warmer. This superfluid 
helium is called helium II. To reduce the flow of this liquid, a constriction is placed 
at the top of the cooling well. However, because of the zero viscosity of liquid helium II, 
a large quantity still passes through the aperture. Large vacuum pumps capable of 
pumping 2500 liters/min at a pressure of 0.1 mm Hg are required to remove the evap¬ 
orated gas. 

12.2.5.2. Special-Construction Dewars. In addition to the cooled-detector-type 
dewar, special dewars have been built that have cooled enclosures and apertures. 
These dewars are the same as an ordinary dewar, except the detector is mounted inside 
the coolant well (Fig. 12-29). The aperture well is coated with an opaque substance 



Fig. 12-29 


Construction of a cooled aperture detector. 






























SPACE-ENVIRONMENT COOLING SYSTEMS 


561 


to ensure that the detector views a relatively cool background. The inside wall of 
the dewar and the face of the aperture plate is usually coated with a material of good 
reflection characteristics to reduce heat leak from radiation. Cooling the detector 
enclosure and the aperture plate decreases their background radiation, and also results 
in significant changes in several detector parameters, as indicated by the investigations 
on PbSe surfaces [15], and on PbS surfaces [16]. 

The contribution of optical system filters to the background radiation can be sig¬ 
nificantly decreased by lowering the temperature of the filter. This can be accom¬ 
plished most conveniently by placing the filter inside the dewar between the window 
and the detector (see Chapter 11). 

12.3. Space-Environment Cooling Systems [17] 

Space cooling systems require extreme reliability, often coupled with a requirement 
for long periods of unattended operation. They are particularly restricted as to size 
and weight. The space environments of high vacuum and zero gravity create problems 
of liquid behavior. Development of a cooling system based on the sublimation of a 
solid coolant into the high vacuum of space shows considerable promise since the 
problems associated with vapor liquid separation while venting in a zero gravity 
field are avoided. This cooling system consists of a solidified gas or liquid, an insulated 
container, an evaporation path to space, and a conduction path from the coolant to 
the device being cooled (Fig. 12-30). The operating temperature obtainable with this 
system depends upon the choice of coolant, the pressure maintained in the system, 
and the heat load. By varying the vapor flow rate, which in turn regulates the back 
pressure and temperature of the effluent flow, a specific operating temperature can be 
maintained. The system’s operating time depends on the amount of coolant and the 
heat load. 



Heat Transfer 
Rod 


Bellows Sealed 
Access Port 


Bellows Sealed Gas 
Escape Port 


Super- 
Insulation 


Supports for 
Inner-Container 

Inner-Container 
Wall 


Outer-Container 

Wall 


Fig. 12-30. Space environment coolant system [17], 


12.3.1. Operating Principles. The space environment cooling system is based 
on the pressure dependency of the temperature of a solid in equilibrium with its vapor. 
The vapor pressure versus temperature plot for methane is shown in Fig. 12-31. For 
an operating temperature of 77° K using solid methane, the vapor pressure must be 
maintained at a value of 9.0 torrs. Addition of heat vaporizes the solid coolant and 
would cause the vapor pressure, and thus the temperature, to increase. The tempera¬ 
ture and pressure are maintained constant by venting this vaporized coolant to the 
































562 


DETECTOR COOLING SYSTEMS 



o 

£ 

s 


w 

K 

D 

% 

w 

K 

a* 

S 

D 

D 

U 

< 

> 



TEMPERATURE (°K) 


Fig. 12-31. Vapor pressure versus temperature of 
solid methane [17]. 


high vacuum of space. The quantity of heat that a given weight of solid coolant re¬ 
moves in vaporizing is equal to the sum of the heat of fusion and the heat of vaporization. 
For example, to sublimate 1 g of solid hydrogen, which has a heat of fusion of 0.0544 
joule/kg and a heat of vaporization of 0.452 joule/kg, 506 joules of heat input are re¬ 
quired. The rate at which the solid vaporizes per unit surface area is given by 

Q= ioo VWf <12 - 31) 


where Q = rate in g sec -1 cm -2 

P = vapor pressure at temperature T in torrs 
T = temperature in ° K 
M = molecular weight of the coolant 

12.3.2. Design. The main factors entering into the design are the choice of coolant, 
surface-to-volume ratio of the container, and insulation requirements of the container 
and support structures. 



















SPACE-ENVIRONMENT COOLING SYSTEMS 


563 


The temperature and pressure ranges for several solid coolants are given in Table 
12-9. Solid hydrogen is most promising because of the wide temperature range avail¬ 
able. Systems have been conceived using solid hydrogen for cooling up to 140° K. 
At a higher temperature of 200° K, solid ammonia is another promising refrigerant 
because of its high density, which results in a smaller and lighter storage container. 


Table 12-9. Temperature and Pressure Ranges 
of Solid Coolants [17] 


Coolant 

Temperature Range 

(°K) 

Pressure Range 
(torrs)* 

Methane 

90-67 

80-1 

Argon 

83-55 

400-1 

Carbon monoxide 

68-51 

10-1 

Nitrogen 

62-47 

73.6-1 

Neon 

24-16 

260-1 

Hydrogen 

14-10 

56-2 


"“Below this pressure only a 1° to 3° temperature drop takes place. 


A cylinder with its length L equal to its diameter D is used as the coolant container 
since it has the lowest surface-to-volume ratio of any simple configuration. Super¬ 
insulation (Table 12-2) having a low thermal conductivity of 0.5 /jlw (cm °K) -1 is used 
as the insulating material. Heat leak through the support structures is reduced by 
using materials of low thermal conductivity and small cross sectional area with the 
greatest possible length. 

The main source of heat in the system is the heat leak into the coolant container. 
This is the critical element in obtaining optimum system weight as a function of oper¬ 
ating time. In particular, the heat leak through the seals must be a minimum. These 
seals surround the heat transfer rod from the detector to the coolant and provide a 
path to the pressure control. Bellows are used as seals instead of straight tubes 
because of the greater heat-path length they provide for the same linear thickness. 
A 2.54 X 10 4 m-thick stainless-steel bellows with a heat-path length five times the 
insulation thickness is considered suitable. 

This design provides a methane cooler with a one-year operating temperature of 
77° K that weighs 13.8 kg, whereas a hydrogen cooler with a one-year operating tem¬ 
perature of 13° K weighs 31.7 kg. If a bellows seal of thinner material (5.08 x 10 -5 m) 
and a heat-path length 20 times the insulation thickness is used, weight reductions 
of 9.06 kg for a methane cooler and 20.2 kg for a hydrogen cooler can be achieved. 
Thus, reducing the heat leak through the insulation or the ports provides a more effi¬ 
cient cooler in which the heat load of the detector is the controlling factor. 

12.3.3. Practical Cooling Systems. The results of calculations of system weight 
using hydrogen, neon, nitrogen, carbon monoxide, argon, and methane are given 
in Table 12-10. Coolers using hydrogen and methane have the lowest weight for 
their respective temperature ranges. System weight and volume as a function of 
operating time for a solid-methane cooler are shown in Fig. 12-32 and 12-33, respec¬ 
tively. System weight and volume as a function of operating time for a solid-hydrogen 


564 


DETECTOR COOLING SYSTEMS 


Table 12-10. Coolant and Insulation Weights 
for Cylindrical Container ( L=D) and 
One-Year Operation [17] 



Coolant 

Coolant 

Temperature 


(°K) 

Hydrogen 

12 

Neon 

24 

Nitrogen 

61 

Carbon monoxide 

68 

Argon 

84 

Methane 

88 


Detector head load =100 mw 
Outer container temperature = 300° K 


Weight of Coolant 
and Insulation 
(kg) 

29.8 

53.9 
28 
26.6 
30 

11.9 



3 6 9 12 

OPERATING TIME (months) 


3 

U 


W 


S 


D 

J 

O 

> 



3 6 9 12 

OPERATING TIME (months) 


Fig. 12-32. System weight as a 
function of operating time at 
various outer container tempera¬ 
tures for methane coolant [17]. 


Fig. 12-33. System volume as a 
function of operating time at 
various outer container tempera¬ 
tures for methane coolant [17]. 





















SPACE-ENVIRONMENT COOLING SYSTEMS 


565 


80 


60 


H 

X 

O 

W 

£ 40 
S 

W 

H 

co 

CO 


20 


Coolant Tem 
Heat Load 1( 
(0.3413 Btu/ 

perature 13°K 

)0 mw 

lr) 

/ 

/ 

/ 

/ 

/ 

540°R (30 
400°R (222 

°° K) X / 

° K) \V' 

7 

A# 

// 

// 

AS 

: 360°R 
(200°K) ^ 

Y 

/ 

/ 

/ 

/ A 

- /Y 
/ // 

Y / 

c< 

/ 1 

Y — 

Y 

s' 

80°R (100°K) 

Y 

Y 

_1_1_ 

_1_1_ 

_1_1_ 


6 9 

OPERATING TIME (months) 


12 



Fig. 12-34. System weight as a func¬ 
tion of operating time at various outer 
container temperatures for hydrogen 
coolant [17]. 


Fig. 12-35. System volume as a func¬ 
tion of operating time at various outer 
container temperatures for hydrogen 
coolant [17], 



Fig. 12-36. Increases of methane cooling system weight with 
decrease in coolant temperature [17], 









































566 


DETECTOR COOLING SYSTEMS 


cooler are shown in Fig. 12-34 and 12-35, respectively. The effects of ambient tempera¬ 
ture dre also indicated on these plots. The weight and volume of the methane and 
hydrogen cooling systems were calculated using only one operating temperature for 
each coolant. Figure 12-36 shows the results of calculations made to determine the 
change in weight of the system when different temperatures are used. The weight of 
the container, the pressure-control, and the heat-transfer rod were neglected in the 
calculations for Fig. 12-36. The container weight does not exceed 0.453 kg for any 
configuration or operating time, and the pressure-control and transfer-rod weights 
were estimated to total 1.812 kg. 

12.4. List of Manufacturers 
Aerojet-General, Azusa, California 

AiResearch Manufacturing Division, Garrett Corp., 9851 S. Sepulveda Boulevard, 
Los Angeles 45, California 

Air Products and Chemicals, Allentown, Pennsylvania 
Arthur D. Little, Acorn Park, Cambridge, Massachusetts 
Cryogenerators, Division N.A. Philips Co., Ashton, Rhode Island 
Energy Conversion Inc., 336 Main Street, Cambridge 42, Massachusetts 
Fairchild Stratos, 1800 Rosecrans Avenue, Manhattan Beach, California 
Hofman Laboratories, Inc., 5 Evans Terminal, Hillside, New Jersey 
Honeywell, Military Products Group, 1915 Armacost Avenue, Los Angeles 25, California 
International Telephone and Telegraph Corp., Industrial Laboratories Division, Fort 
Wayne, Indiana 

Jepson Thermoelectric, Inc., 139 Nevada Street, El Segundo, California 

Linde Co., Div. Union Carbide Corp., 270 Park Avenue, New York 17, New York 

Malaker Laboratories, Inc., West Main Street, High Bridge, New Jersey 

Martin Company, Electronic Systems and Products Division, Baltimore 3, Maryland 

Materials Electronics Products Corp., 990 Spruce Street, Trenton, New Jersey 

Needco of America, Inc., 5770 Andover Avenue, Montreal 9, Quebec, Canada 

Norden, Division United Aircraft Corp., Norwalk, Connecticut 

Ohio Semiconductors, 1205 Chesapeake Avenue, Columbus 12, Ohio 

Pesco Products Division, Borg-Warner Corp., Bedford, Ohio 

Philco, Lansdale Division, Lansdale, Pennsylvania 

Radiation Electronics, Division, Comptometer Corp., Chicago, Illinois 

Raytheon, Spencer Laboratories, Burlington, Massachusetts 

Santa Barbara Research Center, Goleta, California 

Texas Instrument Inc., Dallas, Texas 

Westinghouse, Air Arm Division, Friendship Airport, Baltimore, Maryland 
References 

1. P. R. Barker and W. L. Brown, "Cooling Devices for Infrared Detectors,” in Infrared Quantum 
Detectors, Institute of Science and Technology, The University of Michigan, Ann Arbor, Mich., 
Rept. No. 2389-50-T (July 1961) (CONFIDENTIAL). 

2. Data Sheet, "Cryogenic Data,” Cryogenic Engineering Co., Denver, Colo. 

3. J. R. Kettler and P. L. Rice, Closed Cycle Helium Refrigeration for Cooling Electronic Compo¬ 
nents, AE-1991-R, Rev. 1. AiResearch Manufacturing Division, Garrett Corp., Los Angeles 
(1962). 

4. Instruction Manual for Nitrogen Pressure Generator, Barnes Engineering Co., Stamford, 
Conn. (1961). 


REFERENCES 


567 


5. Sales Brochure, Infrared Cooling, SER-24-1, AiResearch Manufacturing Division, Garrett 
Corp., Los Angeles (1962). 

6. Sales Brochure, Infrared Components, Santa Barbara Research Center, Goleta, California 
(1962). 

7. Sales Brochure, Moisture Trap for Drying High Pressure Nitrogen Gas, ITT Components and 
Instrumentation Lab., Fort Wayne, Ind. (1961). 

8. R. H. Vought, "Peltier Cooling,” in Infrared Quantum Detectors, Institute of Science and 
Technology, The University of Michigan, Ann Arbor, Mich., Rept. No. 2389-50-T (July 1961) 
(CONFIDENTIAL) 

9. B. L. Worsnop, Applications of Thermoelectricity, Wiley, New York, 8-106 (1960). 

10. R. R. Heikes and R. W. Ure, Jr., Thermoelectricity, Science and Engineering, Interscience, 
New York, 15, 458-517 (1961). 

11. Sales Brochure, Thermoelectric Cascade Cooler 094438-010, Pesco Products Division, Borg- 
Warner Corp., Bedford, Ohio (1962). 

12. M. B. Grier, Proc. IRE, 47 (1959). 

13. F. E. Jaumet, Proc. IRE, 46, 538 (1958). 

14. H. J. Goldsmid, Applications of Thermoelectricity, Wiley, New York, N. Y. (1960). 

15. R. M. Talley, T. H. Johnson, and D. E. Bode, Proc. IRIS, 4, No. 4 (1959). 

16. J. J. McArdle, Proc. IRIS, 6, No. 1, 107, Infrared Industries, Waltham, Mass. (1961). 

17. A. I. Weinstein, A. S. Friedman, and U. E. Gross, Proc. IRIS, 7, No. 2,187-191, Aerojet-General 
Corp., Azusa, Calif. (1962). 














































































Chapter 13 

FILM 


Allan L. Sorem 

Eastman Kodak Company 


Gwynn H. Suits 

The University of Michigan 


CONTENTS 


13.1. Introduction. 570 

13.2. Available Infrared Films and Plates. 570 

13.3. Hypersensitizing. 570 

13.4. Definition of Density and Exposure. 573 

13.5. Sensitometric Characteristics. 573 

13.6. Spectral Sensitivity and Filter Transmittances. 574 

13.7. Reciprocity Characteristics. 574 

13.8. References for Additional Details. 574 


569 












13. Film 


13.1. Introduction 

Most of the characteristics of infrared photographic film and plates are not unlike 
those of visible light films and plates. The primary distinction is that the infrared 
emulsion responds to radiation in the near-infrared spectral region as well as to some 
part of the visible and ultraviolet. 

i 

Infrared films can be handled like similar conventional materials, except that extra 
precautions should be observed to make sure that the films will not be fogged by un¬ 
suspected stray infrared radiation. Loading and unloading of cameras, magazines, or 
cassettes should preferably be done in total darkness. Certain woods, hard rubbers, 
plastics, and black papers are not opaque to near infrared radiation. Thus, accidental 
fogging of infrared film can occur when improper protective covering is used for the 
plates. Metal foil on black paper is a dependable covering. If a safe-light is needed 
during processing, it must be used with a filter specifically recommended for infrared 
materials, since most safe-lights transmit infrared freely. The Kodak Safe-Light 
Filter, Wratten Series 7, is recommended. 

When infrared film is used in equipment designed primarily for visible-light photog¬ 
raphy, some additional precautions should be noted. The lens focal positions indicated 
by barrel markings or range finders on cameras designed for visible-light photography 
do not apply accurately when infrared film is used. The longer wavelength rays are 
refracted less by lenses so that the effective focal length of the lens is about 0.5% longer 
than the visible-light focal length. In addition, aberration corrections which are made 
to optimize the sharpness of the image in the visible portion of the spectrum are not 
generally optimum for the infrared portion. Exposure meters designed for visible light 
use do not respond to near-infrared radiation. Hence, these meter readings are not 
useful as exposure indicators unless the ratio of visible to infrared radiation is known 
beforehand. 

13.2. Available Infrared Films and Plates 

Infrared films are available in roll form for still and motion-picture cameras, and in 
sheet form. In addition, spectroscopic plates and films with spectral sensitivity in the 
near-infrared region are available. Table 13-1 lists brief descriptions of these materials 
as an indication of the types of films currently available and of the forms in which they 
are sold through dealers. 

13.3. Hypersensitizing 

The speeds of some infrared films and plates can be increased by hypersensitizing 
them. For best results this should be done just before exposure, and in any case the 
films and plates should be used within a few days after hypersensitizing. This proce¬ 
dure is essential for Kodak spectroscopic films and plates having M, N, and Z sensi- 
tizings. 

Films or plates to be hypersensitized should be bathed in a weak solution of ammonia 
made by diluting 4 parts of 28% ammonia (the strongest available commercially) with 


570 


Table 13-1. Infrared Films and Plates 


HYPERSENSITIZING 


571 


0) G 

|-g 

8 . § 

« JS 
c c 

~ g 

co 

C bo 
O C 

.H 

+3 CO 
03 to 
•p 0> 

3 8 

> c 
a 

-g X 

.tJ c 

>• 3 


u fb A 


s a 

•C -w 

c £ 
a> 0) 

•s £ 

CO ^ 

-I 

O ^3 


to -tj 

& c 
S 10 


,s g 

G *2 

0) 

8 £ 


CO 

$ 2 ) 

8 
O 

a 

5 

o 
X 

o-x 
.. a 
►> 

-c 

- CO c 

— s- CO 


CO 

O 

C 

cO 

A 

3 

£ 

5 

o 


fee 

3 

o 


- -M 

. >. 5- 

A -A O 

a a £ 


g 8 

■ <-* i~> 

"a. 

ax 

tO CD 
- £ 
.2 <3 

I s 

to be 
c 3 

•2 -c 


3 

cr 

0) 

t- 


a 

* g to 

o s-o 
JS -2 fc 
< ^ ° 


CO 

•g 

0) 

a 


CO 

to" 

(O 

9 

8 

a, 


i T3 bO 

] c a 

> co z 

CJ 


1 

o. 


(V 


0) 


5b cc 
2 £ 


•2 ,2 

X 8 

0) 

CO M 

8 g 

8 3 

fe. & 

c *" 

01 CO 

X ’ 

£ w 


« ,o X 
,fh <+_, 

3 b 

bo c 2 

cO a) tfc! 

2 £ .2 

O bo 
8^ 
J Si* 

<21 o 


£1 

c 15 
o) -5 

to a. 
0 ) 

b "O 
a c 

E § 

x 
3 43 


CO 


CO to 

' 3 


X ~ 
m -G 
CO “ 

cO 3 
0 ) A. 

CO _Q 

X (O 
CO 

x t 3 

C 3 

* c 
01 

to to 

-tJ 3 

C E 

*3 

2 o> 

O. 

0) 
bo 


h - >. <• 

8 z o) to 
•£> 2 ^ — 
&<2 


AA 

O CO 


3 

2 

3 

O 

CO 


-X 

cO 

X 

o 

* 

c 

CO 

£ 

CO 

CO 

W 


3 

■g 


CO 

5 co 

tH 

3 u 


£ 3 

"g 


0 ) 

2 3 

T3 

W^ 5 


3 cc 
fa "§ 


3 

(h 

3 

■g 


tH 

Z 


Ol, 


to 

cO 

W 


CO 
Ol 

-C 
8 

a r 
3 o 
P O 


Io 

■g 

X 

£ CO 

CO u 

£ 3 

1 s 


io 

*g 

5 CO 

CO G 

£ -2 

1 s 


«o 


O 

^ N 
9 CO 


3 

'C 


1 

CO 

— CO 

o 

>,.2 
•B 03 

■§ J 
&«= 
a> 

^ 8 

— J3 

< ” 


CO 


be 

cd 

£ 

E 

3 

CO „ 
!«£ 
2 c 
o -R o 

CO cO »ft 

P c5 


^ o 


-2 

I 


o 

o 


T3 

3 

CO 

1^2 to 

.§ 
P 1 to 

« £ 

23 
- 8 
< « 


CO 

a> 

N 

‘S 

E 

I 1 < 

CC 




CO 

o 

* hi 

CO CO 

c ^ 

O) 

t3 "9 
5 e 

^ 0 

-c 

o 


to -2 
CO T 3 
u , 

*> 

c 3 

u 

>,•2 
^ 0) 
0> « N 

■g -3 3 

3 ro <-i 

•8 s § 

s 


c 

o 

•£ 

cc 

u 

-♦-> 

a> 

c 

8. 


fl) 

5S & 

2 CO 
^ 8 

o £ 

x a 

JS 3 

+J 

>>.2 

3 S 

cd o 
u Cp 

•S £ 




-3 ^ 

^3 Sh 
« 3 
-3 O g 

»-i CO N 

O X> 2 
0 > O Xi 

>>•2 

>>'u ® 

JS 3 -g 

&! §• 
g)l £ 
I ® « 

a5 3 


§ J 

C cc 

£ > 
2 3 
in 5P 


ec 


v 

c 

Ol 

Ih 

3 


••= 2 
to S 
to D- 
,3 c 


O 

i- 

CL, 


O 

£ 

£ 

? g 
uo 3 

00 cd 

T3 E 

2 3 
3 -g 


? O 

co Z 


tH tH 

3 3 
o 

j- >» 

»« e 
01 ^ 
> 

3 


Z 43 
< > 


sj 


tH 

a> - 

3 * 

& a5 
8 -3 

c 2 
A E 
H 


0 ) 2 
-a 3 

3 A 
bC m 
3 - 
35 
3 
o 


c £ 
« b 

co CX 
co ^_r 
JZ 
X be 
cj 

2 C 
^ 0) 
^ 0) 


tn 

cd 
co 

_ ^ 
>> 
S 3 


fee 
1*5 

— 3 

3 > 

C 

O 


c 3 3 
3 §> 8 
3 3 2 
be 3 

in 

— 43 Ou 

2 w TJ 

3 c 

— >H 3 

3 3 

>> >> -3 

54 —■ bo 
<D u 
33 "“ 

JrS’g 

T3 O S- 


^3 

^ > 

3 S 

CO 

aT G 
■ - a> 

CO 


be 

cd 

£ 


3 . . 

3 

O be 

5 | 

c 

3 
>. 
o 


3 

G 

3 


3 


»H 

3 

>> 

_ 3 

C —' 

§cT 

£ 3 .£ a 


nH 

T5 0) 

S § 
i"8 



1^ 
.2 £ 


-x 

3 


11 

2 h 

tC 

3 rT 
^ £ 




m 

co 

c 

3 



£ 

£ 

4 ) 

be 

0 

f-H 


CD 

3 

W 

3 

z 

CO 

in 


“ £ 

A ~ 

be ft, 

£ 'S 

irt 3 

’g'a 


•g 

£ o 
3 -3 
1_ -C 

cE Q. 

3 3 

2 fee 

I s| 
|<E 


3 

£ 

a E 

2 43 

3 < 

3 xi 
w 3 

M 3 
3 .Vh 

•g C 


572 


FILM 


CO 

-St 

s. 

0 

s 

a; 


2} 8 

l M 
^ 0> 

P S 

, tt> 

cd w 
CO 

2 <d 
, N 
o> +3 

r* ’55 

£ fi 
H o> 
n 

CL 0) 

ft a 

X 

0) ^ t* 

— a) co 

< ^ ^ 


Q 

W 

c 

OS 

J- 

2 


•l 

3 

O 

CO 


CD 
3 
3 

* 

3 
O 

O 

CO 

p 

H 

< 

J 

Dh 

Q 

Z 

< 

CO 

2 

ri -s 

a 



CO 
CL 
N 

§ CO 


T3 

C 

£0 CO 

£ g 

£^ 

»g 

1 s 

2 6 

CD 


S 

o 

<N 


05 

co o X 

a; ic 
Is X co 

E 00 


£ v 
o X 

05 


Tf 

X 

■'t 


£ 

o 

CD 

H 

X 

CO 


s O , 
co 

x x *7 x 

h n n w 


CO 

r-H 

w 
—1 

CQ 

< 

E- 1 


.8 

CO 

c 

<L 

Ji "3 

2 5 

a ° 

-e 

O 


Hoc 

3 „ © 

y 5“ 

<U Cd 

g 'E.5 

■ u 2 

i fe = 


cfl 

cd 

h 


8 .- 


O <D o 
c a to 

8 be « ® 
• c o o 
g | o ^ 

3 0> 

^ O d > 

T) co C 3 
a) a) —i .3 
| 6<C ® 


3 4* 

■°oi ’> 

5 ' o .-2 

CO 

3 C 

, o « 

’ Ja “ 

—M cd 

>> -* 

E-H Q 3 

-3 a, 

2 K • 

> - 3 . 

u • 33 « ~~ 

3 .d _ 00 
— M d 
3 c .3 0 
C <D C -*j 

in co * « 


1o * 

I a 

§ g> 


o 

uw c/} 
<D 
J-, 


be 

15 


O *7 ■T 

a d 

CO 

g a 

3 

CO 

. CO 

C 3 


c i« 


CD .ST 

£ -= 

£ £ 
3 S 
x > 

W 


3 D 

S 2 g 

cC _ 

*T £ 3 
3 -u -Q 


'S « — 

3 -2 3 <N >> 

E £•&-£ 

- | cd 3 5 

£ § * s 

.£ U S -X 

ji . 3 2 

o c > a 

CO C-H . 

(D ^ .t; co 


CO 

cd 

i-i 
-*-> 

C 

8 

1 

• 2 ^ j; 

"3 C C -t-> 
cd c t* <D -r 
^ 3 o CO £ 


' 1 - 

*-• rt) 
<L> ^ 

! I 

c bo 

-t-T .£ 

i 8 

c id 
o s- 

0 S 

£ — 
s >> 


3. 3 > 

^ S 3 


>>030 

o d ’st 3 

5 C . >, 

2 W gj 

g C S 

« 2® c 
u a ^ o 

° CD ^ * 

s u 3 -X . 

3 3 9 3 3 

"o. 8 3 a e 


a 

o C0 

CJ 0) 


•3 


0 cd 

L-, rr- 1 

c 

CO 

«L 

Q 

■M Ph 

a 

«0 

S.’S 

£ 

0, 

CO 3 

cC 


-* 2 
3 £ 


a 

>> 


a 

>> 

E- 


•3 — 


> 

a 

>> 

H 


N 

I 

a 

>> 

E- 


3 

00 

o 

a 

>> 

E- 1 


SENSITOMETRIC CHARACTERISTICS 


573 


100 parts of water. Concentrated ammonia is a dangerous irritant and should be 
handled cautiously. The diluted ammonia solution should preferably be used at a 
ture of about 40°F but never at a temperature exceeding 55°F. The films or plates 
should be bathed for about 3 minutes. The material should then be bathed for 2 to 3 
minutes in methyl or ethyl alcohol and dried as rapidly as possible in a stream of cool, 
dust-free air from a blower or fan. 

Kodak Spectroscopic Plate and Film Type I-N can be hypersensitized by using plain 
water instead of the ammonia solution described above. 

Care must be exercised in hypersensitizing to prevent streaking. In drying, partic¬ 
ular care must be taken to keep dust from the emulsion surface, to prevent spotting. 

13.4. Definition of Density and Exposure 

The exposure and development of photographic film produces an image consisting 
of areas having different transmittances, depending on the number and size of the 
silver grains present. If the transmittance is measured by the ratio of the intensity 
of the undeviated light passing through the plate or film to that of the incident colli¬ 
mated light on the back, then the transmittance is called the specular transmittance. 
If the transmittance is measured by the ratio of the intensity of the undeviated and 
scattered light together to that of the incident collimated light, then the transmittance 
is called the diffuse transmittance. The opacity, O, is defined as the reciprocal of the 
transmittance. The density, D, is defined as logio O. Thus, depending upon the man¬ 
ner of measurement of transmittance, there is a corresponding specular and diffuse 
density. The density values used in the data of this chapter are diffuse density values. 

The exposure, E, may be expressed either as the time integral of the illuminance, I, 
or the time integral of irradiance, H, so that 


and 


E (photometric) = C f f V(\)H (A) dk dt 

Jo J\i 


E (radiometric) 


id = f f 

■'O A, 


//(A) dX. dt 


where V (A) is the relative visibility curve and C is a conversion factor. In photographic 
literature, when E is expressed as a photometric quantity, the units of E are customarily 
meter-candle-seconds. When E is expressed as a radiometric quantity, the units of 
E are customarily ergs per square centimeter. 

13.5. Sensitometric Characteristics 

The curves presenting density as a function of logio E are known as the H and D 
curves (after Hurter and Driffield), or usually just the characteristic curves (see Fig. 
13-1 to 13-8). The standard source of radiation for obtaining these curves is an artificial 
source which is made to provide an irradiance the spectral distribution of which is close 
to that provided by a 6100°K blackbody in the restricted spectral interval of interest. 
This source is intended to simulate average daylight (sunlight plus skylight). From the 
illuminance from this source one may derive a unique irradiance in the infrared spectral 
region. Although more direct use of E (radiometric) for characteristic curves of infrared 
film makes good sense, E (photometric) has been used in the literature and is a valid 
indicator of the magnitude of E (radiometric). When a filter is specified along with the 
characteristic curves, the effects of the filter losses are implicit in the characteristic 
curves. The filter is considered to be an integral part of the film. 


574 


FILM 


It is important to point out that the characteristic curves are average properties and 
have a meaning in this regard similar to published characteristics of vacuum tubes. In 
addition, these characteristics hold only for the specified conditions of exposure and 
processing. Changes in exposing light quality or in processing will yield a different 
set of characteristic curves. The response of photographic materials to parameter 
variation tends to be nonlinear with almost every parameter. It cannot be too strongly 
stressed that, when photometric or radiometric measurements are to be made, the 
apparatus to be used must be calibrated for photographic response. Extreme care 
must be exercised to obtain reproducible processing conditions. When possible, 
calibrating exposures should be made adjacent to the areas to be measured. A wealth 
of information concerning the use and problems of photographic materials for photom¬ 
etry may be found in the literature of astronomy. 

13.6. Spectral Sensitivity and Filter Transmittances 

The sensitivity of a photographic film to monochromatic light is defined as the recip¬ 
rocal of the monochromatic exposure (in erg cm -2 ) required to produce a stated den¬ 
sity above the fog level. The common logarithm of the sensitivity as a function of 
wavelength is given in Fig. 13-9 through 13-14. Notice that in Fig. 13-12 the sensi¬ 
tivity curve for D = 0.3 is not exactly a vertical translation of the curve for D = 1.0. This 
indicates that gamma is a function of wavelength as well as a function of development 
time. Transmission curves for some useful filters are found in Fig. 13-15. 

13.7. Reciprocity Characteristics 

The reciprocity law states that, when all other parameters are held fixed, the final 
density on the film depends only upon the produce of the illuminance due to the standard 
source and the exposure time. This law does not hold for extremes in exposure condi¬ 
tions. Very high intensity light of short duration and very low intensity light of long 
duration are both less effective in achieving a given density than moderate values of 
light and exposure times. The reciprocity characteristics are shown in Fig. 13-16 
through 13-18, which are plots of log exposure in meter-candle-seconds vs. log illumi¬ 
nance (photometric irradiance), with constant exposure time lines indicated. The 
degree of deviation from a straight horizontal line shows the degree of reciprocity-law 
failure. 

13.8. References for Additional Details 

An important reference for more detailed discussion of infrared photographic mater¬ 
ials, sources of illumination, and applications, is Walter Clark, Photography by Infrared, 
(Wiley, New York, 1946). A good discussion of photographic materials and materials 
and photometric concepts and terminology may be found in Hardy and Perrin, The 
Principles of Optics, (McGraw-Hill, New York, 1932). A chapter devoted to "Astro¬ 
nomical Photographic Photometry” may be found in F. E. Ross, The Physics of the 
Developed Photographic Image (Van Nostrand, New York, 1924). 

Additional information about infrared photographic materials may be found in two 
Kodak publications: Infrared and Ultraviolet Photography, No. M-3, and Kodak Plates 
and Films for Science and Industry, No. P-9. 


FILM 


575 



TIME OF DEVELOPMENT 
(minutes) 



Fig. 13.1. Characteristics of Kodak Infrared Sheet 
Wratten A filter No. 25, developed with intermittent 


Film exposed to sunlight througl 
agitation at 68°F (20°C). 


3.0 


2.0 - 


< 

o 


1.0 


0.0 


Microdol-X 



Time-Gamma Curves 
l l l I I i i i I i i i 


0 10 20 
TIME OF DEVELOPMENT 
(minutes) 


13.0 min y = 1.00 
11.0 min y = 0.90 
9.0 min y = 0.80 

7.5 min y = 0.70 

6.5 min y = 0.80 


Density of 
Antihalation 
Base 

i 



L 


1 


Characteristic 
Curves for D-76 


3.0 


2.0 


- 1.0 


■ 1.00 0.00 1.00 

LOG EXPOSURE 


2.00 


0.0 


Fig. 13.2. Characteristics of Kodak Infrared Sheet Film (35 mm), exposed to sunligh 
through Wratten A filter No. 25, developed with intermittent agitation at 68°F (20°C) 



TIME OF DEVELOPMENT 
(minutes) 



Fig. 13-3. Characteristics of Kodak High-Speed Infrared Film (H5218), exposed to 
sunlight through Kodak Wratten filter No. 25, developed in Kodak Developer D-76 at 
68°F (20°C) with intermittent agitation. 


DENSITY ‘ ^ DENSITY DENSITY 





























576 


FILM 



TIME OF DEVELOPMENT 
(minutes) 



Fig. 13-4. Characteristics of Eastman Infrared Negative Film, Type 5210, exposed to 
daylight through Kodak Wratten filter No. 25, developed in Kodak Developer No. D-76 
in a sensitometric machine. 


Illuminant: Infrared (Daylight 
through Wratten #25 Filter) 
Development: DK-50 at 68°F (20°C) 
in Sensitometric Machine 


Illuminant: Infrared (Daylight 
through Wratten #25 Filter) 
Development: D-19 at 68°F (20°C) 
in Sensitometric Machine 


2.8 

2.4 

2.0 >< 

1 6 « 
1,0 OT 

1.2 g 
0 8 ® 
0.4 
0.0 

- 2.0 - 1.0 0.0 
LOG EXPOSURE 
A 



B 


2.8 
2.4 
2.0 
1.6 
1.2 

0.8 
0.4 
0.0 

0 5 10 15 20 

TIME OF DEVELOPMENT 
(minutes) 





Fig. 13-5. Characteristics of Kodak Infrared Aerographic Film. Daylight through a No. 25 filter 
is the illuminant; for A the developer was DK-50 at 68°F (20°C), for B it was D-19 at 68°F (20°C) — 
both in a sensitometric machine. 



TIME OF DEVELOPMENT 
(minutes) 



>« 

H 

►—i 

w 

z 

w 

Q 


Fig. 13-6. Characteristics of Kodak Spectroscopic Plate and Film, Type I-N hypersensitized 
with water, exposed to daylight through Kodak Wratten filter No. 25, developed in Kodak 
Developer No. D-19 at 68°F (20°F) in a sensitometric machine. 




















FILM 


577 



(minutes) 



LOG EXPOSURE 


Fig. 13-7. Characteristics of Kodak Spectroscopic Plate and Film, Type IV-N, hypersensitized 
with ammonia, exposed to daylight through Kodak Wratten filter No. 25, developed in Kodak 
Developer No. D-19 in a sensitometric machine. 



TIME OF DEVELOPMENT 
(minutes) 



Fig. 13-8. Characteristics of Kodak Spectroscopic Plate and Film, Type IV-N, hypersen¬ 
sitized with ammonia, exposed to daylight through a Kodak Wratten filter No. 25, devel¬ 
oped in Kodak Developer No. D-76 at 68°F (20°C) a sensitometric machine. 












578 


FILM 



Fig. 13-9. Spectral sensitivity of Kodak Infrared Film developed in Kodak 
Developer DK-50 5 min at 68°F (20°F). D — 0.6 above gross fog. 



Fig. 13-10. Spectral sensitivity of Eastman Infrared Negative Film, Type 
5210 developed in Kodak Developer D-76 9 min at 68°F (20°F). D = 0.6 
above gross fog. 






LOG SENSITIVITY 


FILM 


579 



Fig. 13-11. Spectral sensitivity of Kodak High-Speed Infrared Film 
developed in Kodak Developer D-76 7-1/2 min at 67°F (20°F). D = 0.6 
above gross fog. Dashed lines show effective sensitivities when the Kodak 
Wratten filters indicated are used. 


>< 



Fig. 13-12. Spectral sensitivity of Kodak 
Spectroscopic Plate and Film, Type I-N. 




Fig. 13-13. Spectral sensitiv- Fig. 13-14. Spectral sensitivity of Kodak 

ity of Kodak Spectroscopic Plate Spectroscopic Plate and Film, Type I-Z. 

and Film, Type I-M. 











580 


FILM 



Fig. 13-15. Transmittance of Kodak Wratten filters used for 
infrared photography. 


EXPOSURE TIME (sec) 



(meter candles) 

Fig. 13-16. Reciprocity characteristics for exposure to 3000°K tungsten filament through 
Kodak Wratten filter No. 25. Films developed in Kodak Developer D-76 at 68°F (20°C) 
for D — 1.0 above gross fog. A: Kodak Infrared Film. B : Eastman Infrared Film, Type 
5210. C: Kodak High-Speed Infrared Film. 











FILM 


581 


EXPOSURE. TIME (sec) 



Fig. 13-17. Reciprocity characteristics of Kodak Infrared Aerographic Film for D = 1.0 
above gross fog. 


EXPOSURE TIME (sec) 



LOG ILLUMINANCE 
(meter candles) 

Fig. 13-18. Reciprocity characteristics for exposure to 3000°K tungsten filament through 
Kodak Wratten filter No. 25. Films and plates developed in Kodak Developer D-19 at 
68°F (20°C). D = 1.0 above gross fog. Kodak Spectroscopic Plates and Films. A: Type 
I-N, water hypersensitized. B: Type IV-N, ammonia hypersensitized. C: Type I-Z, 
ammonia hypersensitized. D: Type I-M, ammonia hypersensitized. 




































































Chapter 14 

PREAMPLIFIERS AND 
ASSOCIATED CIRCUITS 

Arthur E. Woodward and David Silvermetz 

Servo Corporation of America 


CONTENTS 


14.1. General Requirements. 584 

14.2. Sources of Amplifier Noise. 584 

14.2.1. Thermal Noise. 584 

14.2.2. Shot Noise. 585 

14.2.3. Flicker Current Noise. 586 

14.2.4. Noise in Semiconductors. 586 

14.3. Noise Factors and Noise Figures. 587 

14.3.1. General Relations. 587 

14.3.2. Detector Noise Factor. 588 

14.3.3. Loading Resistance Noise Factor. 589 

14.3.4. Preamplifier Noise Factor. 589 

14.3.5. Overall Noise Factor. 590 

14.4. High-Frequency Compensation. 590 

14.5. Vacuum Tube Amplifiers. 592 

14.5.1. Selection of Vacuum Tubes. 592 

14.5.2. Input Impedance. 593 

14.5.3. High-Frequency Response. 595 

14.5.4. Very-High-Impedance Amplifier. 596 

14.5.5. Power Supply Considerations. 597 

14.6. Transistor Amplifiers. 597 

14.6.1. Noise vs. Frequency. 597 

14.6.2. Noise vs. Source Impedance and Bias Point. 598 

14.6.3. Transistor Bias Stabilization. 598 

14.6.4. Characterization of Transistor Noise. 599 

14.6.5. Relationship of Minimum Noise Factor 

to Transistor Parameters. 602 

14.6.6. Circuit Considerations. 602 

14.6.7. Transformer Coupling. 604 

14.6.8. Simple Low-Noise Preamplifier Design. 605 

14.7. Field Effect Transistors. 606 

14.7.1. Noise Figure. 606 

14.7.2. FET Preamplifier Design. 608 

14.8. Synchronous Detection. 610 

14.9. Grounding Considerations. 611 

14.9.1. Grounding Techniques. 611 

14.9.2. Low-Noise Cable. 611 


583 







































14. Preamplifiers and Associated Circuits 


14.1. General Requirements 

The characteristics of detectors used in infrared systems are detailed in Chapter 11. 
The impedance of these detectors extends from a few ohms to hundreds of megohms, and 
their output is usually in the order of microvolts. 

The primary objective in preamplifier design is to produce an amplifier that will 
increase the detector signal to a level capable of being transmitted over a cable, possibly 
exceeding 10 feet in length, without degrading the signal-to-noise ratio available at the 
detector. The primary requirements of such an amplifier are low noise, high gain, 
low output impedance, large dynamic range, good linearity, and relative freedom from 
microphonics. The amplifier must be compact, since it is usually mounted near the 
detector, and must be carefully shielded to prevent the introduction of unwanted 
signals by stray fields. 

14.2. Sources of Amplifier Noise [1] 

14.2.1. Thermal Noise [2, 3]. The noise spectrum of the mean-square open circuit 
voltage of thermal noise is 

Seif) = 4 kTR (14-1) 

where k = Boltzmann’s constant 

T = absolute temperature (usually °K) 

R = resistive impedance 
The mean-square open-circuit voltage then is 

e* = 4kTR\f (14-2) 

where A f— effective bandwidth 

V= £- [ G(f)df (14-3) 

tro Jo 

Go = maximum gain 
Gif) — gain as a function of frequency 
The power spectrum is 

S p (f ) = kT (14-4) 

Thus R is a white thermal noise source with kT units of power per unit bandwidth. 
The mean-square noise current spectrum of a Norton source can be written 

Siif) = 4kTG (14-5) 

where G = shunt conductance. Both R and G are functions of frequency: 

R = Re [Z s (f)] G = Re [!%/»(/)] (14-6) 

where Re means "real part of” 

Y S h = shunt admittance 


584 


SOURCES OF AMPLIFIER NOISE 


585 


More useful expressions (for multiport functions, reciprocal networks, and other special 
cases) are given in Bennett [1]. 

14.2.2. Shot Noise [4, 5]. Shot noise is caused by the independent random emission 
times of the discrete electrons. The number of electrons with charge emitted in a 
given time interval n has a Poisson distribution. The average noise current is 

i = l = en ( 14 . 7 ) 

For frequencies small compared to the reciprocal of the transit time, the spectral 
density of the mean-square shot-noise current in a temperature-limited diode is 


Si(f) = ne 2 = 2el 


(14-8) 


In a space-charge-limited diode with zero emission velocity, the Langmuir-Child’s 
equation holds: 


• = ^ /2eV /2 
^ 9 \ra/ d? 

where j = current density 


(14-9) 


V = anode voltage 
d = distance from cathode to anode 


eo = free-space permittivity (dielectric constant) 
m = mass of electron 


ixiui v aiiu. 


The Maxwell probability law of emission velocities is probably 

(. muo/kTc ) exp (—mv 0 2 /2kT c ) 0 v 0 

P(v o) = • 

0 otherwise 

where P(v 0 ) = the probability density that an electron will have a velocity v 0 
T c = cathode temperature 

The approximate equivalent of Child’s law, where Ve ^ kT e , is 


(14-10) 


j = (4e 0 /9)(2e/ m) ll2 [(V a ~ V m ) 3l2 /(d - X m ) 2 1 (14-11) 

where V n = potential at the anode 
V m = minimum potential 

The space-charge limited diode with distributed electron velocities behaves as though 
its cathode were at X m , the position of the minimum potential. Then the spectral 
density of anode current fluctuations is written 

Si(f) = 2eiy 2 (14-12) 

where y 2 = the space-charge-smoothing factors. Values for y are given by Thompson 
[6]. Another useful approximation is 

Of) = 2kTg = 2 kT e ffgd (14-13) 

where T e ff can be obtained from Thompson [6] 

gd is the dynamic conductance of the tube 

g is an effective thermal conductance 

A good approximation is 

T eff = 0.644 T c 


(14-14) 




586 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


A space-charge-limited diode can be treated as a conductance g« operating at 0.644 
times the cathode temperature. 

For the triode, , g 044 


s(f) = 2k(^ 

where g m — dynamic transconductance 


cr 




'-[ ,+ ;©T 


d cp = distance from cathode to plate 
d C g = distance from cathode to grid 
For the pentode, 

S(f) = ( i a it/ic*)ei c + ( iJi c ) 2 [ ei c y 2 4- i c 2 8(f)] 

i a = anode current 


(14-15) 


(14-16) 


(14-17) 


i 2 = second grid current 
i c = cathode current 
8(f) = impulse at zero frequency 
y 2 = smoothing factor 

Additional results for multigrid and negative grid tubes, including the effects of 
flicker and partition noise, are given in Bennett [1] in Chapter 4. 

14.2.3. Flicker Current Noise. The mechanism of flicker noise (l// - noise) has not 
been completely clarified. It is characterized by a power spectrum that varies inversely 
with frequency, hence the name 1 If noise. In vacuum tubes, at low and sub-audio 
frequencies, flicker noise can exceed the shot noise by several orders of magnitude. 
If flicker noise is represented by a noise-current generator, then the noise spectrum is 


,2 



(14-18) 


where K is a constant. In many semiconductors, flicker noise is also the dominant 
source of noise at low frequencies. The noise results from a fluctuation in current 
density caused by some modulation mechanism, and can be divided into two components. 
One component causes random changes at the transistor surface; the other causes 
random changes in the leakage path around the transistor junctions (primarily the 
reverse-biased collector junction). 

14.2.4. Noise in Semiconductors. The mean square current noise spectrum can 
be written 


Si(f ) = 4kTG(f) + 2 el 


(14-19) 


The shot noise of a junction transistor can be represented by noise generators as 
shown in Fig. 14-1. 


Vbn = 4 kTZb 

(14-20) 

ien 2 = 4 kTy e (f) —2 ele 

(14-21) 

V\n = lenZe 


I 2 n 2 = icn Otien = 2 ett(l <x)I e &f 

(14-22) 



NOISE FACTORS AND NOISE FIGURES 


587 


Qfi + i 
e zn 



Fig. 14-1. Noise circuit for transistor. 


Generation-recombination (g-r) noise [7] is a bulk property of semiconductor material 
and results from fluctuations in the conductivity produced by carrier-density changes. 
The variation in carrier-density is caused by the random character of generation, 
recombination, and trapping processes. Depending on the magnitude of the current 
noise, g-r noise may be impossible to detect. As a result, the semiconductor exhibits 
current noise at low frequencies and thermal noise at high frequencies. The mag¬ 
nitude of g-r noise is given by 


2/ 2 tA f 

N [1 + (2t r/V) 2 ] 


(14-23) 


where r = electron-hole lifetime 

N = average total number of free electrons 
f = frequency of modulating signal 

14.3. Noise Factors and Noise Figures [5, 8] 

14.3.1. General Relations. The available (incremental) power A P as of a source 
P as is the maximum average power obtainable (see Fig. 14-2): 

A Pas = e7/4 R s = kT s Af (14-24) 




1 “j 


R » ! 
-W\/-(j)- 

-4 }> 

i i 

(: 

l 1 

5 e 0 

1 j t 

| Input 

R -—| Terminals Output) 


1 

1 Terminals. 


X 

X 1 1 

.. r\ 

Signal 

4 f 

Generator 

| 4-Terminal Network | 


Fig. 14-2. Noise relations in a two-terminal 
pair network. 
























588 PREAMPLIFIERS AND ASSOCIATED CIRCUITS 

The effective noise temperature T es is the available noise from the source N as divided 
by kAf: 

T es = Nas/kbf (14-25) 


The standard noise temperature T 0 is 290° K and the relative noise temperature t s is 
given by 

t s = Tes/To (14-26) 

The available power gain G a is: 


G a = PaJPa, = ~ 

it Q 


H((o)Zi 
Zi + Zs 


(14-27) 


The operating noise factor F 0 is the total available noise output power N a o divided by 
the available noise output power from the source alone N a o s : 


F 0 — N aol N aO s 


(14-28) 


The noise power from an ideal generator is kTAf. 

Other expressions are 

(S/N) as 
" (SW)aO 

Nno Na °i 

Tp —— 1 _J_ 1 T a0 _ 1 _i_ 1 

° N a0g G„kT es Af 
Fa — 1 + ~~ (F — 1) 

1 es 
N a 0 j 

F= 1 +- ! - 

GakToAf 

where N tl(ii = internally generated noise power 
S = signal power 
For cascaded stages 1 .... N 


(14-29) 

(14-30) 

(14-31) 

(14-32) 



N 

itf-1 

G a 

= G a 

G a 


1 

* 1 



n-l 


(14-33) 


F 



+ 


N r ,n-l 

s [(f.-D/n 

n =2 m = 1 




(14-34) 

(14-35) 


The noise figure is the noise factor expressed in decibels, 

NF = lOlogio F (14-36) 

14.3.2. Detector Noise Factor. The noise produced at the output of an infrared 
detector is discussed in Chapter 11 . The detector’s noise factor, which is many times 
greater than thermal noise, is found to be 


F d = akTAf/kTAf= a 


(14-37) 


where a is the excess detector noise. 














NOISE FACTORS AND NOISE FIGURES 


589 



14.3.3. Loading Resistance Noise Factor. In the network shown in Fig. 14-3 
loading resistance R L represents all resistance elements shunting the detector. This 
resistance includes all biasing resistances and the input resistance to the preamplifier. 
The detector is represented by a noiseless resistor in series with a signal generator 
and a noise generator. The loading resistance is represented by a noiseless resistor 
in series with the noise generator alone. The available power gain is given by 



/ Rl \ 2 I 4 RRl 
\R + Rl) / R + Rl 

e s *l4R 


(14-38) 


which reduces to 


G = 


Ri 


R + Rl 


Rl + (i* J ' 


. . -+■ 


The noise factor from Rl is then 


F N kTbf _ R 

L GkTbf ( Rl/R + Rl) kTbf Rl 


(14-39) 


(14-40) 


Thus, Rl should be as large as practical. However, a large Rl introduces additional 
problems in the form of shunting capacitance. 

The noise factor of the detector in cascade with loading resistance is given by 


Fol = F d +^~~ = « + 

Ltd At 


(14-41) 


14.3.4. Preamplifier Noise Factor. At low frequencies, vacuum tube noise can 
be represented by an emf in series with the grid having the relation 

emf= (- IkTReAf) 1 ' 2 (14-42) 

where R e q = equivalent noise resistance of the tube. This can be written as [9] 

R eq = R t +blf (14-43) 

The first term of this equation is the shot noise R g . For a triode, 

R s = 2.5 /g m (14-44) 

where g m = transconductance of the tube. The second term 6//'of Eq. (14-43) is 1 If 
noise (flicker noise). For a good tube (/<, = 1 ma), the factor b is of the order of 10 6 ; 
for average tubes, b is about 10 7 ; and for poor tubes it is about 10 8 or 10*. The factor b 





















590 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


is current-dependent and passes through a minimum of about 7 0 = 1 ma. Vacuum 
tubes should therefore be operated below recommended operating conditions for low- 
noise applications. 

In transistors, two noise sources (generally partially correlated) are used to describe 
the equivalent noise input to the amplifier. (A more complete discussion is given in 
Sec. 14.6.) Figure 14-4 shows a preamplifier represented by a noiseless amplifier in 
series with an equivalent noise resistor. The preamplifier noise factor is then 

F a = 1 + R„ (14-45) 


Noiseless 



Fig. 14-4. Preamplifier noise factor network. 


14.3.5. Overall Noise Factor. The overall noise factor (including detector noise, 
loading resistance noise, and preamplifier noise) is 


FDA 


R n n 

a 4- —-h ReqR 

Rl 



The overall signal-to-noise ratio is 


So 1 St 
No F da Ni 


(14-46) 


(14-47) 


14.4. High-Frequency Compensation 

To minimize the loss in the available signal-to-noise ratio of the detector, the re¬ 
sistance in parallel with the detector should be as large as possible. A high source 
impedance requires high-frequency compensation in an amplifier to correct for band¬ 
width loss due to capacitive input loading. Figure 14-5 illustrates the actual input 
circuit at high frequencies where R = Thevenin equivalent of the detector and parallel 
resistances and C = all capacitive loading at amplifier input. 



Fig. 14-5. Preamplifier input circuit at 
high frequencies. 


O 























HIGH-FREQUENCY COMPENSATION 591 

The magnitude of the signal voltage at the grid of the preamplifier is 

e sg = e s /(l + w 2 R 2 C 2 ) 1 ' 2 (14-48) 

The mean-square noise voltage at the grid of the preamplifier due to the thermal noise of 
R is 

e "’ 2 = 4kT 1 1 + Jfi’C’ df (14-49) 

If the amplifier has a response such that 

G = Go (1 + o) 2 C 2 R 2 ) (14-50) 

then 

e s o = (G 0 ) 1/2 c s (14-51) 

and the noise voltage at the output is given by 

— r ft 

e„o 2 = 4 kT RGK 1 + a > 2 R 2 C 2 ) df= 4kTRG 0 Af (14-52) 


Neglecting the shot noise of the first stage, the signal-to-noise ratio at the output is 

e s ole n o = e s l(4kTRAf)' 12 (14-53) 

This is identical to the signal-to-noise ratio obtained at the input in the absence of any 
shunt capacitance. Therefore, by using a properly compensated amplifier where 
G = G 0 (l + cj 2 R 2 C 2 ) the effect of input capacitance is eliminated without affecting the 
signal-to-noise ratio. 

The noise voltage from R eq at the grid of the preamplifier input tube is given by 

e n r = (4kTR eq Afyi 2 (14-54) 

The noise voltage at the output of the compensated amplifier is called peaked-channel 
noise. It is found to be 


which becomes 


CnrO 2 


4kTR eq Go [ * 

Jfi 


(1 + (o 2 R 2 C 2 ) df 


enro 2 = 4nkTR e<l Go [A f+ 4nR 2 C 2 (Af) 3 /3] 


(14-55) 


(14-56) 


where A f = fi — /i. Peaked-channel noise reduces the fine detail of a signal, but it 
can be tolerated better than flat-channel or white noise which, because of its broad 
spectral structure, could completely distort the signal. 

In many instances, the shot noise can be neglected. If the upper limit of the input 
circuit of a dc system is given by 

f 2 = 1/27 tRC (14-57) 

and the amount of compensation is designated as 

m = fa! fi = Aft fi (14-58) 

then 

c nr„ = (Go) 1/2 {4kTR eq Afy> 2 (1 + m 2 /3) 1/2 (14-59) 

The total ouput noise due to thermal noise R and the shot noise of the input tube is given 

by _ _ 


e„t = (e „ 0 2 + e nr o 2 ) 112 


(14-60) 









592 PREAMPLIFIERS AND ASSOCIATED CIRCUITS 

which is the same as 


e nt = ( 4kTAfG <>) 1/2 [R eq (l + m 2 /3) + R) 1 ' 2 (14-61) 

This is shown in the following example where 

R eg = 1 kfl (triode) 

R = 200 kH 
C = 4 pf 
ft = 500 kc 

This produces the relationships 


ft = 0 (14-62) 

u = ( 2ttRC )-» = [2tt(2 X 10 5 )(4 x lO" 12 )]' 1 = 200 kc (14-63) 

m = ft/ft = 500/200 = 2.5 (14-64) 

R eq (1 + m 2 /3) = 1000 [ 1 + (2.5) 2 /3] = 3080 (14-65) 

Comparing this value with the value of R produces 

R »Reg(l + m*/3) (14-66) 

The shot noise can therefore be neglected. The noise voltage then reduces to 

6nr 0 = ( 4kTRAfGo) 1/2 (14-67) 

which is what is obtained when shot noise is neglected. 

14.5. Vacuum Tube Amplifiers 


Low-noise vacuum tubes are useful as preamplifier stages for high-impedance infrared 
detectors because they provide relatively high gain and wide bandwidths with high 
input impedances. Vacuum tube amplifiers have good dynamic range and age charac¬ 
teristics. Their disadvantages are microphonics, high power requirements, and, 
before the development of the Nuvistor and ceramic tubes, their relatively large size. 
Microphonics can be greatly minimized by using low-microphonic tubes and judicious 
mounting techniques. The power requirements, however, are still much greater than 
those of the semiconductors. 

The preamplifier must be mounted near the detector; the number of stages used 
depends upon the gain required, the bandwidth, and the space available. Since most 
detectors are high-impedance sources, they act as constant-current devices whose 
signal-voltage output is a function of the loading resistance. This resistance should 
be high enough that the thermal noise generated by the loading resistance does not 
limit the signal-to-noise ratio available from the detector. In order to prevent a loss 
in high-frequency response, input capacitance effects must be compensated for. 

The design of several preamplifiers used with various detectors is discussed in the 
following paragraphs. They illustrate practical solutions to specific problems and 
do not necessarily represent the best amplifier circuits available. 

14.5.1. Selection of Vacuum Tubes. The major considerations in the selection 
of vacuum tubes for preamplifiers are low noise, low microphonic response, and available 
space. The shot noise of a triode is represented by the equivalent noise resistance 
given in Eq. (14-44). In the pentode, the shot noise can be represented by 


VACUUM TUBE AMPLIFIERS 


593 


R s 


h 

lb “t" Ig2 


2.5 | 20 I g2 \ 

&m &m 2 / 


(14-68) 


where h = average plate current 
I g = average screen current 

Since the shot noise of a pentode is considerably higher than that of a triode, triodes 
should be used in preamplifiers. The calculated equivalent noise resistances of various 
tubes are given in Table 14-1. Vacuum tubes used for low-noise, low-microphonic 
preamplifiers are: 

6112 dual triode 
CK6533WA single triode 
CK8096 single triode 
12AY7 dual triode 


Table 14-1. Equivalent Noise Resistances of Various Tubes 


Tube Type 

§m 

Req 

6AK5 (triode connected) 

6,670 

375 

6F4 triode 

5,800 

430 

417A triode 

24,000 

90 

6J4 triode 

12,000 

210 

6AK5 pentode 

5,000 

1,880 

6AG5 pentode 

5,000 

1,640 

436A pentode 

28,000 

210 

6112 triode 

2,500 

1,000 

CK8096 triode 

1,750 

1,400 

6CW4 triode 

12,500 

200 


14.5.2. Input Impedance. High input impedance in vacuum tube amplifiers can be 
obtained in two ways. The first is to utilize a large grid resistor. Values up to 10 Mfi 
can be used, while values limited to 1 MU or less are not uncommon. The second is 
the use of negative feedback. This method increases the input impedance, stabilizes 
amplifier gain, and improves the low-frequency response. In addition, the use of nega¬ 
tive feedback does not affect the signal-to-noise ratio at the output of the amplifier [10]. 

The use of negative feedback in an amplifier is shown in Fig. 14-6. The closed-loop 
gain is given by 




(14-69) 


where A = open-loop gain of the amplifier. 



Fig. 14-6. Negative feedback in amplifiers. 


















594 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


f3 = Ri(/Rf, for R/>>Rk (14-70) 

The input impedance is given by 

R in = r g {AIK) (14-71) 

The input impedance is determined by the largest grid resistor that can be used and 
the open loop gain available. A loop gain A/3 of 4 or greater is required to provide 
adequate gain stabilization and a useful increase in input resistance. Figure 14-7 

shows a single dual triode used as a low-noise preamplifier and having a bandwidth 

of 1 to 1000 cps. The amplifier constants are 

A = 1500 
R g = 2 MO 
R f = 300 kfl 
R k = 390 O 
K = 300 
R in = 10 Mfi 



Figure 14-8 shows the effect of feedback on the signal-to-noise ratio. A noise voltage 
ej is injected between gain stages of the amplifier. Reflecting this voltage back to the 
signal source as an equivalent noise voltage e'j gives 

e'j = ejIGi (14-72) 

The signal-to-noise ratio at the output of the amplifier without feedback (switch S open) 
is then 

£so _ GiG2e s _ £s 
e n o GiG 2 e'j ej 


(14-73) 































VACUUM TUBE AMPLIFIERS 


595 



Fig. 14-8. Effect of feedback on noise. 


The signal-to-noise ratio at the output of the amplifier with feedback (switch S closed) 
is given by 

e'so _ Ae s /(1 — A(3 ) _ G x e s 

e'.o ~ Ae,Kl - A/3) _ e t (14 ' 74) 

Therefore, the signal-to-noise ratio is independent of the loop gain A, regardless of 
the noise source. 

14.5.3. High-Frequency Response. Wide bandwidths and low detector resistances 
are essential in infrared systems. A low-noise, low-input-capacitance amplifier is 
required in order to obtain optimum performance from the newer detectors. Triode 
amplifiers are seriously limited by the Miller effect which increases the input capaci¬ 
tance by the gain of the stage. Pentodes have low input capacitance, but the screen- 
plate partition noise adds to the problem. The cascode amplifier (see Fig. 14-9) provides 
the low noise characteristic of a triode combined with the gain and stability of a pentode. 
It consists of a grounded-cathode amplifier with the input impedance of a grounded- 
grid stage acting as the plate load. The gain of the grounded-cathode first stage is 
given by [11]. 


e 

s 




Z pk2 







O Gnd 


Fig. 14-9. Cascode amplifier. 






































596 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


G i — 


where 


pk\Z iil { Zpk\ “I" Z a) 

Tp\ “t" Zpk i Z 12I (Z i 2 4“ Zpk 1 ) 

r ' P 2 4- Z L 


Z\2 — 


r P 2 = 


M 2 = 


\x' 2 + 1 


r p 2 


1 4- j(t>C P k2rp2 


^2 


1 4 " j(l)Cplc2^p2 

The gain of the grounded-grid second stage is given by 

(fi '2 4 - 1 ) Zl 


G 2 — 


r ' P 2 + Zi 


The overall gain is 


G — G 1 G 2 


(14-75) 

(14-76) 

(14-77) 

(14-78) 


(14-79) 


(14-80) 


At low frequencies, where Z L = Rl and the interelectrode impedances can be neglected, 
the gain becomes 


G = 


—^l(lX2 + 1)#L 


r p i(fX 2 + 1) + r p 2 + Rl 
Furthermore, if r p i(/x 2 + 1) >> r p2 + Rl, then the gain reduces to 

G ~ g m^R L 


(14-81) 


(14-82) 


The cascode amplifier therefore has the gain of a pentode. 

The gain of the first stage is low as a result of the low effective plate load. Therefore, 
the effective input capacitance is also low. Two problems associated with the use of 
this circuit are (1) the need to provide a fixed bias for the grid of the grounded-grid 
stage, and (2) the need for a high-plate-voltage source to supply the series-connected 
tubes. The noise properties of this amplifier are discussed in [12]. 

14.5.4. Very-High-Impedance Amplifier. The very-high-impedance amplifier shown 
in Fig. 14-10 uses a cascode amplifier with feedback to the input grid to further re¬ 
duce the effect of input capacitance. It is used as a summing or operational amplifier 


Detector 

Bias 



Fig. 14-10. Very-high-impedance amplifier. 













































TRANSISTOR AMPLIFIERS 


597 


e 1 O-WV 



Fig. 14-11. Summing amplifier. 


in analog computers. The equations for its transfer function are developed on the basis 
of the circuit for the summing amplifier shown in Fig. 14-11. The nodal equation for 
the input grid is 

[(e 0 /A)—e,] fl,- 1 4- [(e 0 /A)-e 2 ] R t ~ l + [(e 0 /A-e 0 ] Ro _1 = 0 (14-83) 

By simplifying and rearranging terms, where the feedback coefficient is given by 

(3= [1 + (R 0 /Rx)(Ro/R2)]- 1 (14-84) 

and where A(3 is very large, Ri and R 2 » R 0 , and e 2 is a ground potential, one obtains 

eo/ei = — R 0 /R 1 (14-85) 

Thus, when the amplifier loop gain is very high, the output is independent of the actual 
gain and is a gunction only of the ratio of R 0 to R u 
The input impedance of a summing amplifier is given by 

R in = R 0 /A (14-86) 

As a result, the effects of input capacitance are substantially reduced. However, the 
reduction in input impedance results in a stage gain of approximately unity (if R 0 = 
R 1 ). Therefore, additional stages of low-noise amplification are needed to provide the 
required gain. (See Fig. 14-10 where the first amplifier stage is followed by a low-noise 
amplifier similar to the one shown in Fig. 14-7.) 

14.5.5. Power Supply Considerations. Because of the very low level signals 
amplified by preamplifiers, the noise level introduced through the power supplies should 
be much less than the noise introduced at the input. In most preamplifiers, the noise 
or ripple voltage on the B+ supply should be less than 200 /xv. For the filament supply, 
500 ixv is the maximum. 

14.6. Transistor Amplifiers 

14.6.1. Noise vs. Frequency. The power spectrum of transistor noise vs. frequency 
is given in Fig. 14-12. For typical planar transistors the break point /i occurs between 
100 and 1000 cps. Above /i, shot noise, which has a flat power spectrum, predominates. 
Below /i, the noise figure varies inversely with frequency, with a slope of approximately 
3 db per octave. This region, termed the Ilf region, is dominated by surface and leakage 
noise. 



Fig. 14-12. Transistor noise vs. frequency power spectrum. 




















598 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


At a frequency fa the noise figure begins to increase with a slope of 6 db per octave. 
This increase is caused by a decrease in power gain rather than an increase in internal 
noise. The frequency /i> is given by 

/i“/.*(l-ao) 1/f (14-87) 

where f a b = common base cutoff frequency 
£*o = current gain 

14.6.2. Noise vs. Source Impedance and Bias Point. Figure 14-13 shows the 
broadband noise figure of a low-noise amplifier having a half-power passband of 10 to 
10,000 cps. The noise figure is given as a function of source impedance for three 
different values of emitter current. Figure 14-13 shows that the optimum source 
impedance increases as the emitter current is decreased. 


•Q 

S 

w 

K 

Z> 

O 

HH 

Pm 

w 

a 

o 

z 


9 

O 

K 

ffl 


\Z 



R g , GENERATOR RESISTANCE (kfi) 

Fig. 14-13. Broadband noise figure us. source impedance. 


14.6.3. Transistor Bias Stabilization. 

14.6.3.1. Bias Stabilization and Dynamic Range. Since the noise figure is a function 
of collector current, transistors must be biased for low-noise operation (generally 
between 10 /xa and 200 /xa). As the source impedance increases, the optimum collector 
current usually decreases. The use of the low-noise region is limited by the problems 
of maintaining bias stabilization through fluctuations in ambient temperature and 
changing equipment parameters. However, the biasing problem is greatly reduced 
by using low-noise silicon planar transistors which have low collector leakage currents 
and high current gains. 

A transistor stage cannot accept any signal large enough to cause the collector to 
swing into saturation or cutoff. Drift in the dc collector-to-emitter voltage of a properly 
designed stage lowers this acceptable signal level, thereby lowering the dynamic range 
of the preamplifier. Thus, dynamic range is reduced if bias stability is reduced. 

14.6.3.2. Bias Stabilization Relating to Low-Noise Designs. Transistor stability 
is the ratio of the incremental change in the total collector current, I c , to an incremental 
change in the collector leakage current, I c o, and is given by 








TRANSISTOR AMPLIFIERS 


599 

If is the equivalent dc resistance in the base circuit and Re the equivalent dc re¬ 
sistance in the emitter circuit, then the stability is approximately 


g — ^ B Re 

Rb( 1 + h/b) 4- Re 

For values of hfb almost equal to —1, the expression reduces to 


(14-89) 


S - 


Re ~t~ Rb 

Re 


(14-90) 


In high-impedance RC-coupled amplifiers, Rb is large in order to maintain a high input 
impedance. If R B >> Re, the stability approaches the forward current gain of the 
transistor {Heb)- In order to improve the stability, the emitter resistance Re must be 
increased. However, as Re is increased, the allowable collector swing is decreased. 
Therefore, each design involves compromises between opposing factors. 

Before the development of planar transistors, an I c o of less than 20 na (nanoamperes) 
at 25° C was rare in high-frequency transistors. The 7 c o would double for approxi¬ 
mately every 10° C increase, so that at 85° C, 7 c0 =1.3 /u,a. With a typical stability of 5, 
the change in I c would have been 5.5 fxa, making operation at a nominal collector 
current of 10 /xa (a good low-noise point) impossible. 

The typical collector leakage current of a good low-noise planar transistor is less than 
1 na at 25° C, and doubles with approximately every 14° C increase. At 85° C, with 
Ico less than 52 na, and with a bias stability of 5, the change in 7 C caused by the change 
in Ico would be less than 0.25 /na. In this case, operation at a nominal collector current 
of 10 /na would be feasible. 

In Eq. (14-89), if R B is 1 Mfl and S = 5, then 7? £ must be of the order of 250 kfl (assum¬ 
ing h FB close to unity). The power supply voltage dropped across Re is then 10 /na 
times 250 kfl, or 2.5 volts. If a stability of 3 is desired, R E must increase to approxi¬ 
mately 500 kfl, causing a 5-volt drop across Re- With this large a drop, 6-volt battery 
operation would not be feasible. 

14.6.4. Characterization of Transistor Noise. Although shot and thermal noise 
are predictable from noise theory with a high degree of accuracy, 1 If noise is not. 
Noise figure expressions should therefore be determined by experimental means. The 
noise specified is sometimes measured at a center frequency of 1000 cps, with an effective 
noise bandwidth of 1 cps. For most infrared preamplifier applications, where perform¬ 
ance below 1000 cps is important, such a specification is useless. Much more important 
is the broadband noise figure, which can be considered a noise figure averaged over 
the passband of the amplifier. 

The variation of the noise figure with source impedance is also important. In Fig. 
14-14, an ideal noiseless amplifier, with equivalent noise current and noise voltage 
sources connected to the input, is substituted for a noisy amplifier. The equivalent 
rms input noise voltage e n is found by measuring the output noise voltage with the 




Fig. 14-14. Transistor equivalent noise generator. 

















600 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


input shorted, then dividing by the amplifier voltage gain. The equivalent rms input 
noise current i„ is found by measuring the output noise voltage with the input open, 
then dividing by amplifier voltage gain and input impedance. Since e n and i„ are 
randomly fluctuating quantities, the degree of correlation between them must be known 
before the total power resulting from their combined effect can be computed. The 
degree of correlation is represented by a correlation coefficient, y. 

The noise factor is [13] 


F — 1 + (4 kTB) 1 [(e„ 2 /Rg) + in 2 Rg + 2ye n in\ (14-91) 

The correlation coefficient can range between 0 and 1. Neglecting the 1/ f region, 
y = (h FE )~ 112 for low emitter currents. Thus, for large values of current gain, y can be 
quite small. In the 1 If region, the correlation coefficient increases slightly. 

If Eq. (14-91) is differentiated with respect to R g and set equal to zero, the optimum 
value of R g is found to be 

Ropt = e n lin (14-92) 

Substituting R op t in Eq. (14-91), the minimum noise factor can be found to be 

Fmin = 1 + (1 + y) e n iJ2kTB (14-93) 

The values e n and i„ are functions of I E . Therefore, F is valid only at the bias condition 
at which e n and i„ are measured. These two noise generators are fairly independent 
of collector voltage for voltages below 6 to 10 volts. 

Assuming the noise sources are uncorrelated (y — 0), 

Fmin = 1 + e n inl2kTB (14-94) 


By combining equations, F can be expressed in terms Fmin and R op t as 


F — 1 + (Fmin ~ 1) K 


(14-95) 


where 



Rc 


.R 


opt 



(14-96) 


The value of K may be found in Fig. 14-15. As an example, assume the following 
conditions: 

minimum noise figure = 1.5 db (F m ,„ =1.4) 


R 0 pt = 15 kfl 
R g = 120 kO 



Fig. 14-15. Noise-figure bandwidth. 







NOISE FIGURE (db) 


TRANSISTOR AMPLIFIERS 601 

Since R g /R 0 pt = 8, k is found from Fig. 14-15 to be 4. Therefore, the noise factor is 

F=l + (1.4-1)(4) = 2.6 (14-97) 

and the noise figure NF is 

NF = 10 logio 2.6 = 4.1 db (14-98) 

Minimum noise figure and optimum generator resistance depend upon the frequency 
band to be utilized. Table 14-2 lists data for a typical 2N2176 low-noise silicon alloy 
transistor. Figure 14-16 shows the noise figure of the 2N2176 transistor as a function 
of source resistance for the three amplifier types. Figure 14-17 illustrates e n and i„ 
as functions of emitter current [14]. 

Table 14-2. 2N2176 Low-Noise Silicon Alloy Transistor Data 


Amplifier 

Passband 

B 

e n 

in 

Ropt 

F min* 

NF i min 

0.8 - 50 cps 

80 cps 

0.16 /xv 

40 fxfxa 

4k 

3.0 

4.75 db 

fo = 1 kc 

100 cps 

0.056 /xv 

4.8 fx/xa 

12k 

1.68 

2.25 db 

0.8 - 10 kc 

15 kc 

0.7 

100 fx/xa 

7k 

2.17 

3.3 db 


I c = 20 fx a 
Vce = — 1.5 v 


* Assuming noise generators have been fully correlated (y = 1) for worst-case analysis. 



Fig. 14-16. 2N2176 transistor noise figure vs. source resistance. 



























602 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 



1 jia 10 100 na. 1 ma 10 ma 


EMITTER CURRENT 

Fig. 14-17. 2N2176 transistor emitter current vs. equivalent input noise voltage and current. 


14.6.5. Relationship of Minimum Noise Factor to Transistor Parameters. The 

minimum noise factor F m ,„ can be related to transistor parameters in the following 
manner. In the white noise region, it can be shown [15] that 

- 4kTB(r e l2 + r' b ) (14-99) 

and 

77 * 2e(I c /h FE )B (14-100) 


where r e = ac emitter resistance 

r'b = base spreading resistance 

Since r e = kTlqI E and I E == Ic, substitution into Eq. (14-94) yields 


min 


* 1 + 


1 + (2e/kT)I c r'bV 12 


iFE 


(14-101) 


Therefore, as h FE decreases or I c increases, the noise increases. 

14.6.6. Circuit Considerations. In infrared systems having a high signal-to-noise 
ratio, preamplifier noise factors of 10 to 20 db are acceptable, and preamplifier design 
is straightforward. However, in systems with low signal-to-noise ratios, a low-noise 
preamplifier is mandatory. 

Minimum noise factor and optimum source impedance remain the same for the 
common emitter, common collector, or common base configurations. The common 
collector is often used in high-impedance designs because of its high input impedance. 
However, the common emitter is most desirable because of its greater available power 
gain. High power gain in the first stage is essential if noise from successive stages 
is to be minimized. The following factors should be considered in low-noise designs: 









TRANSISTOR AMPLIFIERS 603 

(1) A transistor specifically designed for low-noise and low-leakage current should be 
used. 

(2) Because Ilf semiconductor noise is a major contributor to overall noise, the am¬ 
plifier’s low-frequency half-power point should be no lower than required by system 
considerations. 

(3) The noise figure depends largely on emitter current (essentially collector current) 
and source impedance. In general, bias current for optimum operation occurs between 
10 and 300 /xa. However, plotting noise figure vs source impedance for a specific value 
of emitter current results in a rather broad minimum. Table 14-3 indicates the noise 
figure minima for modern low-noise planar transistors. For a source impedance far 
below 500 ohms, the best minimum noise figure is obtained by using a step-up trans¬ 
former at the input stage. This method also provides additional voltage gain. Very 
low collector current is required if the minimum noise figure is obtained using a source 
impedance in the megohm region. Bias stabilization becomes difficult, so the transistor 
must have low collector leakage current if operation over a wide temperature range is 
required. 


Table 14-3. Noise Figure Minima 
for Planar Transistors 


Rg 


Ie 

(ohm) 


(/aa) 

0.5-3k 


300 

1.5 — 8 k 


100 

3k-15k 


30 

8k-40k 


10 

15k-200k 


3 

100 k — 2 m 


1 


( 4 ) The fact that the a cutoff frequency of a transistor decreases with emitter current 
should also be considered. The narrowband noise figure starts rising at f=fab (1— ao) 1/2 . 
Therefore, it is important to use a transistor with a cutoff frequency much higher than 
the frequency desired. At low emitter (or collector) current, requirements for low-noise 
figure and the desired frequency response can necessitate the use of a higher bias current 
value. As an example, Table 14-4 lists data for the 2N2645 low-noise, high-gain, 
planar transistor having a 10-volt collector-to-base voltage. From the relationship 
1 —a 0 ~ 1 lh FE , the noise figure is found to rise at 13.5 kc for I c = 1 fx a; at 95 kc for I c = 
10 fjia] and at 560 kc for I c = 100 /x a. Therefore, in the design of a 50-kc low-noise am¬ 
plifier, the first stage would not be operated at a bias current of 1 fxa. 


Table 14-4. Data for 2N2645 
Planar Transistor 

Ic 


1 fx a 
hf'E'. 35 

fab : 80 kc 


10 fia 

55 

700 kc 


100 /xa 
80 

5 Me 



604 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


(5) The bias point should be stabilized against temperature variations (Sec. 14.6.3). 
A stability factor S of 5 or less is usually necessary. 

(6) The optimum noise figure of the transistor is always increased by the effect 
of the base divider network and any unbypassed emitter resistance. This effect is 
negligible if the following conditions are satisfied: 

R b 2* 10 R g R E ^ 0.1 R g (14-102) 

where Re = the external unbypassed emitter resistance. The minimum noise figure 
and the optimum source impedance of a transistor are essentially independent of any 
applied feedback. They are affected only by the circuit components which must be 
added to apply the feedback (such as an unbypassed emitter resistor) and their presence 
can only increase the noise figure. The effect of associated circuitry on noise perform¬ 
ance is discussed in [16]. 

(7) Care should be taken when using Zener diodes in dc coupling and low-level 
biasing. Zener diodes generate large amounts of noise which can be reduced by by¬ 
passing the diode with a large capacitor; however, the Zener diode is not recommended 
for use in a low-noise-input stage which must respond to low frequencies. 

14.6.7. Transformer Coupling. In many cases of preamplifier design, transformers 
provide an excellent solution to coupling problems presented by low- and high-im- 
pedance sources. For low-impedance sources, a step-up transformer can reduce 
amplifier stages and complexity. For high-impedance sources, a step-down transformer 
can be useful, provided the ratio between required high- and low-frequency response 
is no more than a few decades. Broadband transformer design becomes increasingly 
difficult at higher impedance levels. The lower the frequency to be passed, the larger 
the transformer must become to provide (a) the required magnetizing inductance, and 
(b) the necessary dynamic range. Enough iron must be used to maintain B max well 
below saturation level for the largest input signal. The limiting factor is usually 
the magnetizing inductance required. 


R 

g 



n 2 R 


L 


Fig. 14-18. Low-frequency equivalent circuit 
of a transformer. 


The low-frequency equivalent circuit of a transformer is shown in Fig. 14-18. It 
is assumed that 


r p < R g r s < R L 


and the low-frequency 3-db point is given by 












TRANSISTOR AMPLIFIERS 


605 

Assuming that a close-to-optimum noise figure can be obtained by impedance matching 
a 1-Mfi detector to a preamplifier having a 10 kfl input impedance, and that a one- 
cycle low-frequency response is desired, then 

n 2 R L = R c = 1 Mfl 


f i = 1 cps 

Solving for the required primary inductance 



10 6 X 10« 
2tt(1)(2 X 10 6 ) 


= 8000 


hy 


(14-104) 


Such a value is unreasonable. However, for a 1000-cps response, a realistic 8 henries 
is required. 

Transformers usually have appreciable stray capacitance because of the great number 
of windings over a small area. This limits the high-frequency response. Input 
transformers must operate at very low power levels; consequently, they should be 
enclosed in a magentic shield, especially if the preamplifier is located close to a chopper 
motor or power transformer. The transformer should also contain an electrostatic 
(Faraday) shield between primary and secondary windings to minimize stray capaci¬ 
tance coupling between windings. The inclusion of such a shield, however, increases 
the transformer shunt capacitance, which, again, lowers the high-frequency response. 
Transformer windings should be mechanically rigid to prevent microphonics caused 
by minute capacitance changes. Careful encapsulation will generally reduce trans¬ 
former microphonics to a negligible level. 


14.6.8. Simple Low-Noise Preamplifier Design [17]. The amplifier shown in 
Fig. 14-19 has a voltage gain of approximately 20 db with a flat response from 10 cps 
to 50 kc. By referring to Table 14-3, it can be seen that for R g = 300 kfl, and a bias 
current of about 1 /xa, a low-noise figure is obtained. The collector current of the 
input transistor is 1.5 /xa. The common emitter stage is biased for optimum noise 
performance and provides good power gain. The input impedance is 1.2 Mfl up to 


Input Impedance 



Fig. 14-19. High-impedance wideband amplifier with 2N2484 transistors. 



























606 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


4 kc, and the narrowband noise figure is 1.6 db at 1 kc. The stability factor is ap¬ 
proximately 1.5 and is given by 


S — 1 + Rb/Re 


(14-105) 


where Rb = 3.75 MO 
Re = 7.5 MO 

The characteristics of typical low-noise transistors are listed in Table 14-5. 


Table 14-5. Characteristics of Typical Low-Noise Transistors 


Type 

Wideband* 
Noise Figure 
NF 

Collector Leakage 
Current 

Icbo 

dc Current 

Gain, Hfe 


Additional Data 

2N2484 

1.8 db typ 

3 db max 

10 na max 
at Vcb = 45v 

30 min at /<• = 1 pa 

100 min at I c = 10 pa 

200 min at I c = 500 /xa 

Hfe 

= 15 min at 

F = 1 me, for I c = 50 pa 

2N2645 

3.5 db max 

0.4 na typ 

10 na max 
at Vcb = 60v 

20 min at I c = 10 ga 

60 min at I r = 100 pa 

hfe 

fab 

= 2.5 min at F = 20 me, 
for I c = 10 ma 
= 0.7 me (typical) 
for 7 = 10 pa 

2N930 

4 db max 

2 na max 

100 min at I c — 10 pa 

150 min at I c = 500 pa 

hfe 

= 1 min at F = 30 me, 
for I c = 500 pa 

2N2586 

1.5 db typt 

2 na max 

at V C b = 45v 

80 min at I r = 1 pa 

120 min at I c = 10 pa 

150 min at I c = 500 pa 

hfe 

= 1.5 min at F = 30 me, 
for I c = 500 pa 

2N2524 

2 db typ 

3 db max 

2 na max 

1 na typ 
at V CB = 45v 

60 min at I c = 1 pa 

100 min at I c = 10 pa 

150 min at I c = 500 pa 

hfe 

= 1.5 min at F = 30 me, 
for I c = 500 pa 


* Power bandwidth of 15.7 kc, 3-db po ints at 10 cps and 10 kc, V (B = 5 v, I c — 10 pa, R g = 10 k. 
tNarrowband data indicate the same NF is obtained for I c = 1 pa, R g = 1 Meg. 


14.7. Field-Effect Transistors 

14.7.1. Noise Figure. Although good noise figures can be obtained by using con¬ 
ventional planar transistors with source impedances up to 1 Mfl, at higher impedances 
better results can be obtained with the field-effect transistor (FET). In controlled 
FET’s, the 1 If comer frequency is less than 100 cps, almost an order of magnitude 
lower than most conventional transistors. Figure 14-20 shows the noise figure of 



Fig. 14-20. Field effect transistor noise figure 
us. frequency [18]. 




































FIELD-EFFECT TRANSISTORS 


607 


several FET’s as a function of frequency [18]. The FET provides lower noise figures 
in the low-frequency region, and permits low-noise designs for source impedances 
well into the megohm region. 

Figure 14-14 shows how FET noise can be characterized by an equivalent noise 
voltage generator and an equivalent noise current generator. In the case of the FET, 
i n is very small, leading to a high optimum source impedance. The narrowband 
values of these equivalent generators are shown in Fig. 14-21 as a function of frequency, 
for a low-noise 2N2500 FET. The variation of optimum R g and optimum NF vs. 
frequency appears in Fig. 14-22. Stability can be obtained in FET’s by using self bias. 



FREQUENCY (kc) 

Fig. 14-21. Equivalent input noise voltage and equivalent input 
noise current us. frequency [18]. 



FREQUENCY (kc) 

Fig. 14-22. Optimum noise figure and optimum 
generator resistance vs. frequency [19]. 

































608 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


14.7.2. FET Preamplifier Design. Figure 14-23 shows that excellent results for 
high-impedance preamplifiers can be obtained by combining the FET with a planar 
transistor in a cascode configuration. The Miller effect is reduced in this arrangement, 
thereby increasing the bandwidth of a low-noise preamplifier. In addition, independent 
adjustment of the transistor’s operating conditions permits optimum noise performance. 

Figure 14-24 shows the application of two FET’s in a low-noise, high-impedance 
amplifier [19]. The frequency response of the amplifier is shown in Fig. 14-25, and 
the broadband noise figure vs. source impedance is given in Fig. 14-26. The amplifier 
has a fixed gain of 100 and an input impedance of 30 MO shunted by 8 pf. The broad¬ 
band noise factor is less than 3 db with a generator resistance of 50 kfl to 5 MO. The 
2N2498 FET is operated at a drain current of 1 ma, and the 2N930 transistor is operated 
at I c = 100 /X a. Each bias current is optimum for the respective transistor. The 
second stage is operated common base (cascode connection). The optimum source 
impedance is the same for common base or common emitter. The 10 kH optimum 
source impedance for the second stage is provided by paralleling the 20 kfl emitter 
resistor with the 20 kO drain resistor. 



11 left 1.0 /if 



Fig. 14-24. FET low-noise, high-impedance amplifier [19]. 








































































BROADBAND NOISE FIGURE (db) AMPLIFIER GAIN (db) 


FIELD EFFECT TRANSISTORS 


609 



FREQUENCY (kc) 

Fig. 14-25. Frequency response of FET amplifier [19]. 



0.01 


GENERATOR RESISTANCE (MR) 

Fig. 14-26. FET broadband noise figure vs. source impedance [19]. 






























610 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


14.8. Synchronous Detection 

Radiometric instruments frequently use choppers to interrupt the radiation peri¬ 
odically in the optical path. Choppers permit operation in the high-frequency region 
where detector and amplifier characteristics are better. The frequency of the ampli¬ 
tude-modulated signal produced by the chopper is determined by the chopping rate. 
If the chopper blade is shaped to produce sine wave modulation, the detector output 
will be 

e s = j E s (t) sin out (14-106) 

where E s (t ) = the peak-to-peak signal amplitude 

cu = angular frequency produced by chopper 

A synchronous detector is essentially a narrowband detection system in which the 
target signal is beat with a reference signal of the same frequency producing a dc 
output. A block diagram of a synchronous detector is shown in Fig. 14-27. The output 
of the target detector is an intelligence signal which is amplified and then multiplied 
by a reference signal. The reference signal is generated by a magentic or photoelectric 
device synchronized by the chopper. Therefore, the frequency and phase of the refer¬ 
ence signal and the intelligence signal are the same. The output of the reference 
detector is then 

e r = E r sin cut (14-107) 


Target Detector 



Reference Detector 

Fig. 14-27. Block diagram of synchronous detector. 


The product of the two signals appearing at the output of the multiplier is therefore 

e„,(t) = KE s (t)E r sin 2 cut (14-108) 

or 

e m (t) = KEsit)Er (cos 2a >t — 1) (14-109) 

The passband of the filter is made much less than 2a> to eliminate the unwanted fre¬ 
quency component. The output of the filter is then 

KE 

e 0 it) =~y~ E s {t) (14-110) 

Thus the original intelligence has been recovered. 

The similarity between synchronous detection and cross correlation is discussed 
in [20]. The improvement in the signal-to-noise ratio obtained by synchronous recti¬ 
fication depends upon (a) the form and frequency of the target signal, (b) the form of 





















REFERENCES 


611 


the reference signal, and (c) the power spectrum of the noise entering the correlator. 
If one assumes that the radiation and the reference signals vary sinusoidally with 
time and that the system suffers from band-limited white noise, a figure of merit Q(T) 
can be derived by determining the ratio of the signal-to-noise ratio at the output of 
the filter to the signal-to-noise ratio at the input to the preamplifier. 

14.9. Grounding Considerations 

14.9.1. Grounding Techniques. The grounding of preamplifiers and associated 
circuits is of major importance. Improper grounding can cause self-sustained oscilla¬ 
tions, gain distortions, and numerous other undesirable effects. The following ground¬ 
ing techniques should be followed in all design considerations: 

(1) Ground the amplifier circuit to the chassis at the point of lowest signal level. 

(2) Avoid ground loops. Avoid grounding shielded leads at both ends. Tie the 
power supply return to the amplifier at the point of lowest signal level. 

(3) Do not use a single ground bus for multiple-stage amplifiers; instead, provide a 
ground bus for every amplifier stage having a gain of 100 or more. Return grounds 
of all stages to the point of lowest signal level. If the ground for the highest level 
stages cannot be returned to this point, then a separate power supply return should 
be provided. 

(4) Use large diameter wire for ground returns. For low-level stages, the minimum 
size should be No. 22 or No. 20; for higher level stages, No. 18 is the minimum size. 

(5) Separate the ground leads of low-level and high-level stages. If necessary, 
shield the high level ground lead, then ground the shield at the point of lowest signal 
level. 

(6) Keep currents out of all shields by tying the shield to the ground at the point 
of lowest signal level. 

14.9.2. Low-Noise Cable. A major problem in the design of low-noise preamplifiers 
is the spurious audio frequency noise generated in coaxial cables due to shock, excita¬ 
tion, or vibration. This cable noise can completely mask the desired signal, particularly 
in high-impedance circuits subjected to shock or vibration. The mechanism of noise 
in coaxial cables is discussed in [21]. Miniature low-noise cable is now available 
commercially from Microdot, Inc., under the trade name of Mininoise Coax Cable [21]. 

References 

1. W. R. Bennett, Electrical Noise, McGraw-Hill Book Co., New York, 1960. 

2. J. B. Johnson, Phys. Rev. 32, 97 (1928). 

3. H. Nyquist, Phys. Rev. 33, 110 (1928). 

4. W. Schottky, Ann. Phys. 57, 541 (1918). 

5. W. B. Davenport and W. R. Root, An Introduction to the Theory of Random Signals and Noise, 
McGraw-Hill Co., New York, 1958. 

6. B. J. Thompson, D. O. North, and W. A. Harris, RCA Rev. 1, 269 (1940). 

7. K. M. van Vliet, Proc. IRE 46, 1004 (1958). 

8. S. Goldman, Frequency Analysis Modulation and Noise, McGraw-Hill Book Co., New York, 
1948. 

9. A. van der Zeil, Proc. Nat. Electron. Conf. 17, 454 (1961). 

10. G. E. Valley, Jr., and H. Wallman, Vacuum Tube Amplifiers, Radiation Laboratory Series 
Vol. 18, McGraw-Hill Book Co., New York, 1948. 

11. G. M. Glasford, Fundamentals of Television Engineering, McGraw-Hill Book Co., New York, 
1955. 

12. H. Wallman et al., Proc. IRE 36, 700 (1948). 

13. H. F. Cook, Proc. IRE 50, 2520 (1962). 

14. The Evaluation of Transistor Low-Frequency Noise Characteristics, Sperry Semiconductor 
Technical Application Bulletin No. 2110, Sperry Gyroscope Co., Great Neck, L.I., New York. 


612 


PREAMPLIFIERS AND ASSOCIATED CIRCUITS 


15. N. H. Martens, Solid State Design, 3, 35 (1962). 

16. R. D. Middlebrook and C. A. Mead, Semiconductor Products, 2, 26 (1959). 

17. Fairchild Semiconductor Application, Bull. No. 112, Fairchild Semiconductor Corp., Mountain 
View, Calif. 

18. Low Noise Seminar Papers, Texas Instruments Incorporated, Dallas, Texas, 1963. 

19. Field Effect Transistor Theory and Application, Texas Instruments Incorporated, Dallas, 
Texas. 

20. T. J. Wieting, "Correlation Techniques for Infrared Detection Systems,” Proc. IRIS, V, 3, 57 
(1960). 

21. Coaxial Cable Review, Microdot, Inc., 220 Pasadena Ave., South Pasadena, Calif. 


Chapter 15 
OPTICAL 

FREQUENCY-RESPONSE 

TECHNIQUES 

R. Barakat 

Itek Corporation 


CONTENTS 


15.1. Optical Systems and Linear-System Theory. 614 

15.1.1. General Concepts of Linear-System Theory. 614 

15.1.2. Necessary Formulas of Optical Diffraction Theory. 615 

15.1.3. Transfer Functions of Optical Systems in Incoherent Light ... 621 

15.2. Resolution and Its Ramifications. 627 

15.2.1. Resolution Criteria for Point Sources. 627 

15.2.2. Sine-Wave Resolution. 628 

15.3. Effect of Aberrations on Transfer Function. 629 

15.3.1. Spherical Aberration and Coma. 629 

15.3.2. Computation of Transfer Function of an Actual System. 634 

15.4. Effect of Apodization on the Transfer Function. 636 

15.4.1. Altering High- or Low-Frequency Response by Apodization . . . 636 

15.4.2. Luneberg Apodization Problems. 636 

15.5. Merit Factors. 637 

15.5.1. Linfoot Quality Factors. 637 

15.5.2. Plane of Best Focus. 638 

15.6. Frequency Response Calculations from Lens Design Data. 640 

15.6.1. Computer Calculations of Frequency Response. 640 

15.7. Optical Response Measuring Equipment. 643 


613 





















15. Optical Frequency-Response Techniques 


15.1. Optical Systems and Linear-System Theory 

In large measure the optical terminology of linear-system theory is a transcription 
of the corresponding electrical terminology with minor modifications. Even though 
optical systems are two dimensional and operate in the spatial domain in contrast to 
electrical systems which are generally one dimensional in the time domain, the basic 
analysis is not altered. 

15.1.1. General Concepts of Linear-System Theory. In a linear system the 
relation between the input y,(£) and output yoit) is 

yo = £[y,] (15-1) 

where <£ is an operator characterizing the system and t represents spatial coordinates. 
A system is linear if the following conditions are satisfied: 

(1) Commutative condition: if y* and z, are two inputs, then 

£[yi + zi] = £ [zi + y* ] (15-2) 

(2) Superposition condition: 

£ [y« + Zi] = <£ [y«] + it Oi] = y 0 + z 0 (15-3) 

(3) Proportionality condition: If a is a constant (possibly complex), then 

Oy<] = <x£ [y f ] = e*y 0 (15-4) 

If h is the impulse response of the system, the superposition integral is 

yoit) = f h(t', t)yi(t') dt (15-5) 

J — x 

If the system is stationary (time invariant), then 

yoU + r) =£[yiU + r)] (15-6) 

In the spatial domain such a system is called spatially invariant or isoplanatic. When 
stationary conditions are imposed the superposition integral is 

yo(0 =1 hit — r)y,(r) dr (15-7) 

" — oo 

Denoting by capital letters the Fourier transforms of Eq. (15-7), 

Yoif) = Hif)Y,if) (15-8) 

where f is the frequency (either time or spatial) and Hif) is the transfer function. The 
system can be specified by the impulse response function (in the time domain) or the 
transfer function in the frequency domain. 


614 


OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


615 


15.1.2. Necessary Formulas of Optical Diffraction Theory. 

Let us define two functions: 


a(x,y) = complex amplitude of diffracted image 
t(x,y) = point spread function 

where (:r,y) = coordinates in receiving plane. a(x,y) is not an observable quantity in 
the optical and infrared regions (at least by present techniques); however, its absolute 
square t{x,y) is observable: 

t{x,y) = |aU,y)| 2 (15-9) 

or 

i\x,y) = Ct(x,y) (15-10) 

where i\x,y) is the illuminance (flux per unit area). Since t(x,y) is dimensionless, the 
constant C has the necessary dimensions. 

The normalized illuminance ratio is 


i(x,y) 


t(x,y) 
t A .( 0,0) 

l 


(15-11) 


where ^.(0,0) denotes the illuminance at x = y = 0 (geometric center of diffraction 
pattern) for an aberration-free Airy objective with no losses. The i(x,y) expresses the 
distribution of illuminance in the diffracted image as a function of that at the center 
of the aberration-free Airy aperture and is a ratio (no dimensions). In practice i(x,y) 
is called the illuminance (or intensity). 

15.1.2.1. Kirchhoff Diffraction Theory. Kirchhoff diffraction theory states that the 
complex amplitude due to a point source lying on the optical axis of the system is 


a(x, y) 



A(£, rj) exp 


i y(£x + 7 ) y) 


d£ dr) 


(15-12) 


where k = 2i r/A. = wave number of incident light 


f = focal length of the system 
£, r) = rectangular coordinates in exit pupil plane 

The quantity A(£,r}) or its equivalent A{(3y) is defined below. The infinite range of 
integration is for formal simplicity; actually the region of integration is the area of the 
aperture. In normalized coordinates (where X and Y are dimensionless in the receiving 
plane), 


X = 


2ttixo 

A 


x; 



(15-13) 


where /x 0 and v 0 are the aperture semi-angles in the respective directions. The dimen¬ 
sionless angular coordinates in the aperture are 

£ u 7) v 

P = — = —; y = — = ~ (15-14) 

/Ao / /Xo V{)f V() 

where /x and v are angular coordinates and the distance f is large enough so that the 
tangent of an angle can be approximated by the angle. In dimensionless coordi¬ 
nates, 

a(X, Y)= [ f A((3, y) e W x+ v Y) dp dy 


( 15 - 15 ) 








616 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


15.1.2.2. Pupil Function. The quantity A{ft,y) is termed the pupil (or transmission) 
function and 


A(/3, y) = 


amplitude distribution of wave over exit pupil 
amplitude distribution of wave over entrance pupil 


In the general case the pupil function is complex. A(/3,y ) < 1 for physically realizable 
systems and must vanish outside the aperture. 


A(\ 3, y) = A 0 ((3, y) e i2rrWi0 ' y) {(3, y) is in the aperture 

= 0 (/3, y) is not in the aperture) 


(15-16) 


where A 0 (/3,y) = the real amplitude distribution over exit pupil 

W(f3,y) = the wave-front aberration in wavelength units. The wave-front 
aberration is also called optical difference, aberration function, 
and eikonal. 


Fourier transform theory applied to Eq. (15-15) yields 


A(/8, y) 



a(X,Y) e~ i2 ”W +yY) dXdY 


(15-17) 


This equation is integrated over the entire Fraunhofer receiving plane. Thus the 
complex amplitude a{X,Y) in the Fraunhofer receiving plane is the Fourier transform 
of the pupil function A(/3, y). A knowledge of either a(X,Y ) or A(/3,y) is sufficient to 
compute the other, but a knowledge of only t(X,Y) is not sufficient since phase informa¬ 
tion is lost. There are an infinite number of pupil functions having the same spread 
function. The evaluation of a( X, Y) for an aperture of arbitrary shape is very involved; 
however, for circular and rectangular apertures expressions are available for a{X,Y). 

15.1.2.3. Circular Aperture. For a circular aperture 

a(v ) = [ A(p)J a (vp)p dp (15-18) 

Jo 


where p = 
r 2 = 

r,„ = 

v = 

z = 

f= 

Jo = 


rlr m 

((3 2 + y 2 )f 2 

radius of aperture 

277^,, . 2i7Tr m z 

— S m a = — • - 

distance in receiving plane from center 
focal length 

Bessel function of order zero 


In the special case of an aberration-free Airy objective with no losses, A(p) = 1 and 
straightforward integration yields 

i(p) — [2J\(v)lv] 2 (15-19) 

This is the familiar Airy pattern of physical optics (Fig. 15-1). 

15.1.2.4. Annular and Annulus Apertures. The annular aperture is a circular 
aperture in which the center is blocked out; the annulus aperture has a ring-shaped ob¬ 
struction (Fig. 15-2). The annular aperture is a special case of the annulus aperture 
where e' is 0. 





Illuminance 


OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


617 



Lateral Displacement 

Fig. 15-1. Spread function for circular and slit apertures 
in Fraunhofer receiving plane. 



Fig. 15-2. Geometry of the annular and 
annulus apertures. 










618 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 

The effect of the annular aperture is to decrease the radius of the first minimum of the 
diffraction pattern from its value v = 3.832. The extent to which the minimum is 
decreased depends upon the amount of blocking of the central region of the aperture. 
The resultant gain in resolving power is obtained at a considerable loss of intensity in 
the diffraction pattern. A further constraint is the increased intensity of the secondary 
maxima as the central obstruction is increased. See [1] for a comprehensive treatment 
of the aberration-free annular aperture. Both annular and annulus apertures pos¬ 
sessing spherical aberration are covered in detail [2]. Figure 15-3 illustrates the 
variation in distribution of illuminance in the Fraunhofer receiving plane for various 
aberration-free annular apertures. 



Fig. 15-3. Distribution of illuminance for 
aberration-free annular aperture in Fraunhofer 
receiving plane. 



Fig. 15-4. Straubel pupil function for 
values of n = 0, 1, 2. 


15.1.2.5. Straubel Pupil Function for Circular Aperture. A case of interest, because 
it admits of exact integration, is an aberration-free system with a pupil function of 
the form 


A{p) = (1 — p 2 )” (w = 0,l,2,...) 


(15-20) 


For n = 1 the pupil function follows a parabolic law roughly equivalent to a cosine 
distribution; for n = 2, the A(p ) is approximately a cosine squared distribution. The 
behavior of these pupil functions is shown in Fig. (15-4). The illuminance ratio is 


i(u) = 


2 n+1 n! 


J n +i(v ) 

i >" +1 


2 


(n = 0,1,2,...) 


(15-21) 


The Straubel curves include the Airy system {A — 1) as a subcase. The illuminance 
curves are shown in Fig. 15-5. To emphasize the differences in the patterns the maxi¬ 
mum values of each pattern have been normalized to unity to permit comparison. 
Increasing n broadens out the pattern and suppresses the secondary maxima. 

15.1.2.6. Defocusing. The effect of defocusing away from the Fraunhofer receiving 
plane is given by a term of the form 


gi(u/2)p t — gi(27r/X)W'jp2 


(15-22) 









OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


619 



Fig. 15-5. Distribution of illuminance for 
Straubel pupil function; n — 0, 1, 2. 


where W 2 is the defocusing coefficient measured in wavelength units, and u is the 
dimensionless defocusing parameter. In the special case of an aberration-free circular 
aperture the distribution of illuminance is symmetric about W 2 = 0. The resulting 
point spread function is not expressible in simple form but approaches exist which are 
summarized by Barakat [3]. The main effect of defocusing is to broaden the patterns. 

15.1.2.7. Rectangular and Slit Apertures. The complex amplitude in the Fraunhofer 
receiving plane for a rectangular aperture is 


a{X,Y) = 



A(fi) 



A(y) e iyY dy 


(15-23) 


A special case of the rectangular aperture is the slit aperture, a very narrow rectangular 
aperture where variations in the Y direction can be neglected. Therefore, 


a(X) = 



AQ3) a** dp 


(15-24) 


The center of the aperture is taken to be zero. If the limits of the aperture are (±6/2), 
then 

2 77 " UqX 7T bx 


k 




(15-25) 






620 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


The lossless, aberration-free Airy slit aperture, A((3) = 1, provides the illuminance 


HX) = 



(15-26) 


plotted in Fig. 15-1. The first zero of the slit aperture spread function is smaller than 
that of the corresponding circular aperture. 

15.1.2.8. Variable Pupil Functions for Slit Aperture. Two pupil functions which 
can be integrated exactly are 


A,(/3) = a + (1 - a) cos ^0 (15-27) 

(0 < a *£ 1) 

A 2 (l3) = (1 - a) + a(3 2 (15-28) 

The behavior of the functions is shown in Fig. 15-6 and 15-7. Section 15.4 on apodiza- 
tion contains a discussion of additional pupil functions. 



Fig. 15-6. Pupil function for slit aperture of 
form Ai(/3) = a + (1 — a) [cos (7r/2)]/3. 



Fig. 15-7. Pupil function for slit aperture 
of form A 2 (/3) = (1 — a) + a/3 2 . 







OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


621 


15.1.2.9. Strehl Criterion. The Strehl criterion (Strehl definition, Strehl intensity 
ratio) is not a physically measurable quantity but is a common theoretical measure 
of lens performance. The Strehl criterion (SC) is defined as the ratio of the central 
illuminance (X = Y = 0) of the system under consideration (with the possible inclusion 
of aberrations and variable pupil function) to the central illuminance of an aberration- 
free Airy objective; thus 


SC = i(0,0, A, WO 


#0,0, A, WQ 
t A{ ( 0,0) 


(15-29) 


where A and W are included as parameters to indicate the dependence of the SC on them. 
The SC is unity for perfect systems. The first Luneberg apodization theorem states 
that the Strehl criterion is a maximum for an Airy objective (A = 1); any variable pupil 
function must decrease the Strehl criterion. 

For the slit-aperture pupil functions given by Eq. (15-27) and (15-28), the SC ex¬ 
pressions for the respective pupil functions yield: 


ii(0) = 1 - 



(15-30) 


*2(0) = 




(15-31) 


The decrease in i(0) is quite appreciable for the limiting cases. If the system is aber¬ 
ration free, a decrease in the SC implies that the light from the central maximum must 
end up in the secondary maxima of the diffraction pattern because the system is normal¬ 
ized to constant flux. 

15.1.3. Transfer Functions of Optical Systems in Incoherent Light. If o(o\8) is 

the distribution of intensity in the object plane, i(X,Y) is the distribution of illuminance 
in the image plane, and t{X,Y) is interpreted as the impulse response of the optical 
system in the spatial domain, then by the linearity hypothesis 


i(X, Y) = f f t(X, a; Y, 8)o(a, 8) da d8 

J —an d —on 


(15-32) 


which is the two-dimensional generalization of Eq. (15-5). Assuming spatial invariance, 


i(X,Y) = J J t(X—a; Y—8)o(a, 8) da d8 


(15-33) 


— 00 —00 


A region with the property t(X,a; Y,8) = t(X—a; Y—8 ) is an isoplanatic region. 

Equation 15-33 is in the form of a convolution integral whose Fourier transforms obey 
the product theorem 

7(ci>X» ejy) ~ T((Ox, Q)y)0( Ct>x, COy) (15-34) 


where T(<o x ,a) y ) = transfer function (frequency response) of system 
O(o) x ,( 0 y) = spatial spectrum of object intensity distribution 
/(cox, (o y ) = spatial spectrum of image illuminance distribution 
o>x, o) y = spatial frequencies in x and y directions (radians) 



622 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


Therefore, 


OX, Y) = r 

J —00 < 

i{o) x ,oi y ) exp [ i(o> x X -1- o) y Y )] dio x do) y 

J —00 

(15-35) 

7(o>jr, Oiy) = 1 

J —00 ' 

f i(X,Y ) exp [— i(o) x X + w^Y)] dX dY 

/ — oo 

(15-36) 


and similarly for the other Fourier transform pairs. 

The fundamental equation, (15-34), is in the frequency domain and is the generaliza¬ 
tion of the classical result of monochromatic linear system theory: 


T(a) x °, (i)y°) = 


Output 

Input 


which is obtained by making 0(oi x ,oi y ), I(oi x ,oi y ) behave as delta functions: 


(15-37) 


0 (o) x , Wj) — 8 ( 0 ) x OJ x ^)8(Oi y Oi y °) 

I(0) X , lOy) = 8(Oi X — OJ X °)8((O y — (Jjy°) 


(15-38) 

(15-39) 


where (o x ° and o) y ° are constants. 

In terms of the pupil function a(X,Y): 


T((l) X , Oiy ) 


1 

no, o) 



A((3, y)A*(f3—co x ; y~(o y ) d(i dy 


(15-40) 


7\0,0)= r \A(/3,y)\ 2 d/3dy 

J — 00 J —00 

where 7X0,0) is a normalizing constant such that \T(a> x , Wj,)! ^1. By this relationship 
the transfer function is determined by a knowledge of the pupil function, which in itself 
is a function of the design data of the system ( i.e refractive indices, radii of curvature). 
The basic theorem is that the (incoherent) transfer function is the convolution of the 
pupil function across the exit pupil of the system. The geometry is illustrated in Fig. 
15-8. This is the most practical way of determining the transfer function. Except for 
the simplest cases, the alternative method of computing T{(o x ,(o y ) by evaluating the 
Fourier transform of t(X,Y ) is too complicated. 



Fig. 15-8. Geometry of convolution integral 
required for evaluation of transfer function. 






OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


623 


In the absence of any aberration (W = 0) and any apodization (A = 1), the transfer 
function reduces to the ratio of two areas. This allows one to compute the transfer 
function of a perfect Airy system with an arbitrary aperture by analog methods or a 
planimeter. 

At some point (w' x ,w' y ) the convolved area will be zero and T(to x ,to y ) remains zero 
for any values exceeding < o' x ,(o'y . An optical system acts as a low-pass filter in the 
spatial domain, and has a sharp cutoff. The convolved area is a maximum for zero 
frequency (co x = aj y = 0) and is normalized to unity for convenience. 

The slope of the transfer function at the origin is independent of the presence of 
aberrations. This has the important effect of lessening the influence of aberrations 
at low spatial frequencies. 

In general, T(to x ,to u ) is a complex quantity 

7W, toy) = \T(to x , toy)\ e ie (15-41) 

In the special case where the system is aberration free and has a real pupil function, the 
phase angle 6 is zero and there is no phase shift; i.e., T(to x ,to y ) is real. Furthermore, 
the transfer function is real for any symmetric aberration (defocusing, spherical aberra¬ 
tion). Only asymmetric aberrations lead to complex-valued transfer functions. 

15.1.3.1. Transfer Function as Contrast Ratio. Define image contrast by: 


y-, I max l min 


Ci = - 

Imax "t" Imin 

(15-42) 

A sine wave test target has an intensity distribution of the form 


O(to x 0 , 0) = A 0 + B 0 cos (aj/X) 

(15-43) 

The spatial periodicity is taken in one direction only; A 0 and B 0 
object contrast is 

are constants. The 

•o O max Omin B o 

L° — 

O max + Omin Ao 

(15-44) 

Thus 


0(to x °, 0) = A 0 [l + Co cos (a>x°A)] 

(15-45) 

and hence 


7(a> x °, 0) = B 0 [l + T(oj x °, 0)C o cos (o» x °X)] 

(15-46) 


Thus, the modulus of the transfer function is a measure of the ratio of the image contrast 
to the object contrast. The maximum value of the transfer function is at zero frequency 
(dc response), and it is standard procedure to adopt the normalization that T(to x ,to y ) = 1 
at the origin, co x = to y = 0. The phase 6 of the transfer function is a measure of the 
amount of lateral displacement of the image from the geometric center. It is possible 
to measure 6 experimentally but not as easily as measuring the modulus. A linear 
phase shift translates the diffraction image and thus has no detrimental effect. Non¬ 
linear phase shifts introduce harmonic distortion which, of course, results in serious 
deterioration of the image. For example, coma introduces a cubic phase shift. 


15.1.3.2. Strehl Criterion (SC) Expressed in Terms of Transfer Function. 


SC = 


*(0, 0) 

T a .( 0, 0) 


I I toy) d(0.r d(x)y 

J — oo J— oo 


-OC .x 


I TA ,((Oxi My) d(Ox d(t)y 

J —oo J —oc 


(15-47) 








624 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


upon setting X = Y = 0 in Eq. (15-33). The SC is the area under the transfer function. 
This is an alternative method of obtaining the SC and is convenient if only the transfer 
function is known. The maximum SC (unity) is given by an aberration-free Airy objec¬ 
tive (A = 1). Thus, the area under the transfer function of an actual system can be 
greater than that of a perfect system over a limited range; however, the overall response 
cannot be greater. 

A commonly accepted tolerance condition is that the SC shall not fall below 0.8. A 
system that satisfies this criterion gives an image which is only slightly inferior to that 
of a perfect system. 

15.1.3.3. Transfer Function Expressions. For a slit aperture of length b and a 
constant pupil function 


Tin) = JL f A</3)A*</3 - «,) dj8 = f dp (15-48) 

I (UJ J -1 i w 


The integration is only over the intersection and the pupil function is constant, and o> x 
must be normalized to as is /3, then 


T ( ft ) = ——(1 — ft ) 
7X0) 


(15-49) 


Here 


n = 


(1) X 

2/Lt0 


277 ‘fjr 

b/2f 


4nF 

Ax 


— 2w r F 


where F is the focal ratio and f x = 1/A X . 

When O = 1, the transfer function vanishes and the cutoff frequency is 

io x ' = 1/2 F 


(15-50) 


(15-51) 


15.1.3.4. Transfer Function for Circular Aperture. For a circular aperture of radius 
b with a constant pupil function, 


T(Cl) 



cos- 1 n-nu - a 2 ) 1 ' 2 


(o ^ n ss i) 


(15-52) 


where fl is given by Eq. (15-50) with the diameter of the circle replacing the length of 
the slit. The transfer function for the circular aperture is not a straight line like that of 
the slit aperture. Figure 15-9 shows the behavior of the transfer functions for slit and 
circular apertures. Table 15-1 gives the values for T(H) for different values of O 
(0 H 2? 1). 

15.1.3.5. Transfer Function for Annular and Annulus Apertures. The curves shown 
in Fig. 15-10 are adapted from O’Neill [4], As e the obscuration ratio increases, the 
low-frequency response decreases with a corresponding increase in the high-frequency 
response. In fact, as € approaches unity, the transfer function (being the convolution 
of two very thin rings) will have a spike of height 1 at the origin and a spike of height 
1/2 at O = 1. See also [5]. 

The transfer function for the corresponding annulus aperture is illustrated in Fig. 
15-11. Whereas the annular aperture emphasizes the high frequencies, the annulus 
aperture emphasizes (by proper adjustment of € and e') the intermediate-frequency 
region. 

15.1.3.6. Transfer Function of a Typical Reflecting System. When a catadioptric 
system is used for image formation, the distribution of illuminance of the system takes 










Transfer Function 


OPTICAL SYSTEMS AND LINEAR-SYSTEM THEORY 


625 



Fig. 15-9. Transfer functions for aberration-free slit and circular apertures. 


Table 15-1. Transfer Function for 
Aberration-Free Circular Aperture 


a 

T((l) 

a 

T( a) 

0 

1.0000 

0.55 

0.3368 

0.05 

0.9364 

0.60 

0.2848 

0.10 

0.8729 

0.65 

0.2351 

0.15 

0.8097 

0.70 

0.1881 

0.20 

0.7471 

0.75 

0.1443 

0.25 

0.6850 

0.80 

0.1041 

0.30 

0.6238 

0.85 

0.0681 

0.35 

0.5636 

0.90 

0.0374 

0.40 

0.5046 

0.95 

0.0133 

0.45 

0.4470 

1.00 

0 

0.50 

0.3910 




G 

o 

o 

G 

£ 

u 

<D 

CO 

5 

u 

H 



Spatial Frequency 

Fig. 15-10. Transfer function of annular 
aperture for different amounts of central 
obscuration [6]. 










626 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 



Spatial Frequency 

Fig. 15-11. Transfer function of annulus aperture for different ring widths. 



Fig. 15-12. Transfer function of a typical reflecting system as 
seen off-axis. 


an extremely complex form due to the obscuration of the aperture, and is decidedly 
off-axis. The maximum response will occur when there are no aberrations; this is the 
true aperture limit of the system because aberrations only lower the response. Figure 
15-12 shows the off-axis response of a typical catadioptric system. The transfer func¬ 
tion is now a function of two spatial frequencies since symmetry no longer exists. 

15.1.3.7. Coherent Illumination. Coherently illuminated situations do not usually 
appear in nature and can only be achieved in microscopy or specially designed laboratory 
equipment. The basic equation is of the form of Eq. (15-34) where the object and image 
spatial spectra are now of amplitude rather than of illuminance (amplitude squared). 
The amplitude transfer function relating the output (image) amplitude spatial spec¬ 
trum to the input (object) amplitude spatial spectrum is the pupil function of the system. 
In Fig. 15-13, the transfer function of a slit aperture is shown for both coherent (axial 
illumination) and incoherent light. By displacing the point source off-axis (in the 
coherent case) it is possible to increase the resolution for periodic structures (Abbe’s 
theorem); see [7] and [8]. 

15.1.3.8. Cascaded Systems. So far only the transfer function of the optical sys¬ 
tem has been employed; no provision has been made for including the effect of film or 








RESOLUTION AND ITS RAMIFICATIONS 


627 



Fig. 15-13. Transfer function of aberration- 
free slit aperture for coherent and incoherent 
illumination. 


detectors on the image. If the Fourier approach is used, the transfer functions of 
cascaded systems multiply in frequency space. Thus, the atmosphere’s transfer 
function is 7\, that of the lens is T 2 , and that of the film or detector is T 3 . Now Eq. 
(15-34) generalizes to 

(Oy ) = Tl(a) x , COy)T 2 ((x)x, (Oy)T 3 ((0 X , (t)y)0(0)x, O) y) (15-53) 

If we neglect the atmosphere transfer function, the film or detector transfer function can 
materially alter the aerial image as obtained by the lens. The transfer function of 
films varies widely, but the use of a lens-film combination involves no new principles. 
The importance lies in the transfer function of the optical system. Unfortunately the 
transfer functions of individual lenses do not multiply to give the overall transfer 
function of the lens system, as they are now defined. 

15.2. Resolution and Its Ramifications 

Although resolution criteria exist for special situations, these criteria can only be 
interpreted v/ithin their limited context. Two criteria (Rayleigh and Sparrow) have 
been formulated specifically for dealing with point sources. The third criterion (sine- 
wave resolution) is for situations where sine-wave (in intensity) targets are viewed. Al¬ 
though other situations exist, these criteria are valuable provided that proper caution 
is exercised in their use and interpretation. 

15.2.1. Resolution Criteria for Point Sources. 

15.2.1.1. Rayleigh Resolution Criterion. The Rayleigh criterion states that two 
point sources are resolvable when the maximum of the illuminance produced by the 
first point source falls on the minimum of the illuminance produced by the second point 
source. The Rayleigh criterion is tacitly based upon two assumptions that severely 
restrict the generalization of the criterion: (a) point sources are incoherent; (6) point 
sources are of equal intensity. The two (incoherent) point sources are to be placed a 
distance 8 from the center line of the optical system (the distance between the points 
is 28). When the critical value of 28 is reached the value of 28 is the (Rayleigh) limit 
of resolution, called 8 0 . According to the Rayleigh criterion, the limit of resolution is 





628 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


given by 8 0 = 3.832 for the circular aperture and 8 0 = 3.142 for the slit aperture. Thus 
a slit aperture whose length is equal to the diameter of a circular aperture has a smaller 
Rayleigh resolution (approximately 18%) than that of the circular aperture. In most 
cases the loss of light in using the slit aperture is enough to nullify the gain in resolution. 

15.2.1.2. Sparrow Resolution Criterion. Sparrow notes that at the Rayleigh limit 
there is still a central minimum which lies below the adjacent maxima of the diffraction 
pattern. As the distance between the point sources is further decreased the central 
minimum will become shallower and finally disappear. Sparrow defines the limit of 
resolution as that value of 28 (again defined by 8 0 ) for which this phenomenon takes 
place. As the resultant distribution of illuminance is symmetric about the origin, all 
the odd derivatives with respect to the lateral-displacement parameter vanish at the 
origin, and the analytical statement of the Sparrow resolution criterion is that the 
second derivative of the total distribution of illuminance vanishes on-axis; that is 

d 2 

—i(v,8) = 0 (u = 0) (15-54) 

dir 

which states that at zero visibility ( v ) the resultant distribution of illuminance under¬ 
goes no change in slope. The Sparrow limit of resolution is given by the solution of this 
equation. Unlike the Rayleigh criterion, the Sparrow criterion can be applied to 
sources whose intensities are not equal. In addition, the Sparrow criterion depends 
upon the coherence of the source. The resolution limits for circular and slit apertures 
are listed in Table 15-2. The coherent values of 8 0 are larger than the corresponding 
values for incoherent illumination. 


Table 15-2. Resolution Limits for 
Airy-Type Objectives with Point 
Sources of Equal Intensity [4] 


Circular 


Sparrow Coherent 8 0 = 4.600 
Rayleigh 8 0 = 3.832 

Sparrow Incoherent 8 0 = 2.976 


Slit 

8 0 = 4.164 
8 0 = 3.142 
8 0 = 2.606 


It is also possible to formulate the Sparrow criterion in the spatial frequency domain. 
For example the slit aperture case with incoherent point sources leads to 

[ Cl 2 T(Cl, W,A ) cos (8 0 O) dCl = 0 (15-55) 

•'o 

where T is the transfer function (with the possible inclusion of aberrations W and 
variable pupil function A). Solutions of this equation yield the incoherent Sparrow 
limit 8 0 . The quadratic dependence of O in Eq. (15-55) implies that the high-frequency 
components of the spatial frequency domain are the determining factors for the Sparrow 
incoherent criterion. Thus, any transfer function which enhances the high-frequency 
region (such as the annular aperture) will decrease 8 0 . 

15.2.2. Sine-Wave Resolution. With an incoherently illuminated sine-wave target 
of spatial frequency oj x ° in place of the point-source objects, the distribution of illumi¬ 
nance in the image plane is 

I((o x \ 0) = D' o{l + T(o>A 0) [exp id(u> x °)] C 0 cos ((o x °X)} 


(15-56) 


EFFECT OF ABERRATIONS ON TRANSFER FUNCTION 


629 


The diffraction image of the sine-wave target will possess no gradation of illuminance 
when the cosine term of Eq. (15-56) vanishes, that is, when 

< o x °X = A/2 (15-57) 

which leads to a sine-wave cutoff frequency given by Eq. (15-51). This is the aperture- 
limited resolution of the system and is the maximum value that the particular system 
can have. 

15.2.2.1. Spurious Resolution. When the transfer function vanishes, the illumi¬ 
nance in the image is uniform for spatial frequencies smaller than the cutoff frequency. 
The transfer functions discussed in Section 15.1 are for aberration-free systems. The 
inclusion of aberrations will cause the transfer function in many instances to assume 
the form given in Fig. 15-14, which is the transfer function for a slit aperture with one 
wave of third-order spherical aberration in the marginal receiving plane. This curve 
is typical of what is met in practice. The transfer function has regions where it takes 
on negative values. However, since the transfer function represents contrast, which is 
a positive quantity, negative values are interpreted as arising from a phase shift of 
amount A. This implies that, in the regions where the transfer function is negative, 
black and white lines reverse their original position. This effect is called spurious 
resolution and is very serious because it sets an upper limit on the spatial frequencies 
which are useful. 



Fig. 15-14. Transfer function illustrating spurious resolution. 


15.3. Effect of Aberrations on Transfer Function 

15.3.1. Spherical Aberration and Coma. If an optical system is perfect, the inci¬ 
dent spherical wave front must emerge as a spherical wave front after passing through 
the system. Assume that the system is rotationally symmetric, i.e., has a circular 
aperture. The deviation from the ideal spherical wave front is measured in terms of 
the aberration function W. When W is identically zero, the wave-front aberrations 
vanish and the wave front is spherical. Wave theory of aberrations is covered ex¬ 
tensively in [9], [10], and [11]. 

The aberration function for a circular aperture depends upon p, c f> (polar coordinates 
in the exit pupil), and also upon a (the normalized field variable). When the point 







630 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


source illuminating the aperture is on the axis of the optical system, a vanishes. Be¬ 
cause of rotational symmetry, the variables p, </>, and a in W can only occur in the 
combinations p 2 , a 2 , and pa cos </>. The aberration function has an expansion of the 
form 

W(p, ()>, a) = W 2 op 2 + W 2 op 4 + 1VF31 up 3 cos </> + 2W 2 op 2 a 2 

(15-58) 

+ 2 W 22 P 2 cos 2 </> + 3 Wna 3 p cos (f) + higher-order terms 

where (following Hopkins notation [10] the various W coefficients represent the Seidel 
or third-order aberrations. The first term (W 2 0 = W 2 ) represents longitudinal focal 
shift (defocusing) and is not an actual aberration. The Seidel aberrations are 

(a) W 4 o = W 4 = spherical aberration 

( b ) 1 W 31 = coma 

(c) 2 W 20 = image curvature 

( d) 2 W 22 = astigmatism 

( e ) 3 Wn — distortion 

All of these coefficients are in wavelength units. Spherical aberration is indepen¬ 
dent of a and does not vanish for a point source on-axis. The third-order coma term is 
of great importance as it largely determines the quality of the diffraction image in the 
outer parts of the field. Coma can be regarded as spherical aberration for object points 
lying off the optical axis of the system. Spherical aberration and coma are the most 
serious Seidel aberrations in the sense that they are limiting. If the system has a poor 
transfer function on-axis (due to spherical aberration) then it will be worse off-axis. 
Coma is the limiting aberration for off-axis points. 

The expansion of the aberration function is actually an infinite series in p,<f>,a, and 
the only justification for truncating the series is the tacit assumption that the aperture 
and field are small enough so that powers of these variables higher than the fourth 
can be neglected. In many practical systems such is not the case and the fifth-order 
aberrations must be taken into account to make the analysis of the system realistic. 
Analysis of the single aberrations is a necessary step in the development of transfer- 
function theory as applied to actual systems. See [ 6 ] and [10]. 

15.3.1.1. Defocusing. When W 2 (the defocusing coefficient measured in wavelength 
units) is zero, then the image is located in the paraxial receiving plane. The transfer 
function is an even function of W 2 . The transfer-function curves for small amounts of 
defocusing are shown in Fig. 15-15. Note the very rapid deterioration of the contrast 
as the defocusing coefficient is increased. Spurious resolution appears for even as small 
an amount of defocusing as one wave. 

The transfer function for the rectangular aperture suffering defocusing can be eval¬ 
uated explicitly: 

T(fl x , ft„) = R((l x )T((l y ) (15-59) 

where 

T(fl) = (1 - ft) sine [ 87 rW 2 n(l - ft)] ( 0 ^ 0 ^ 1 ) (15-60) 

The defocused transfer function for a slit aperture (very narrow rectangular aperture) 
is shown in Fig. 15-16 for the same values of W 2 as those corresponding to the circular 
aperture. The chief difference (for corresponding values of W 2 ) is that the changes in 
the slit-aperture transfer function are more pronounced than those for the circular 
aperture. This is a general rule for all aberrations and is due to the smoothing action 


EFFECT OF ABERRATIONS ON TRANSFER FUNCTION 


631 



Spatial Frequency 

Fig. 15-15. Transfer function of circular aperture for various 
amounts of defocusing. 



Fig. 15-16. Transfer function of slit aperture for various 
amounts of defocusing. 


of the convolution integral which is more pronounced in two dimensions than in one 
dimension. See [12] and [13]. 

15.3.1.2. Spherical Aberration. A major effort has gone into determining the effects 
of various orders of spherical aberration on the transfer function. Unfortunately, the 
circular aperture is not amenable to analytic methods because of its complexity, and 
numerical techniques are necessary; see [14]. The Gauss quadrature method was 
employed by Barakat [12] for the case where the aberration function included terms up 
to thirteenth-order spherical aberration: 


W 2n , 0 p 2n 

n= 1 


(15-61) 


For the present, consider the terms W 20 , W 40 , and Wro only. 








632 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


In dealing with third-order spherical aberration, the inclusion of defocusing can 
materially increase the response of the system. Thus, taking the aberration function 
in the form w = W ^ A _ 2 ^ 2) (15-62) 


where n = — W 2 /2W 4 , by varying n. we can shift to any desired receiving plane: 

fx = 0 = paraxial receiving plane 

H = 1/2 = central receiving plane 
1 = marginal receiving plane 

In Fig. 15-17 and 15-18 the transfer functions for a circular aperture are shown for 
a half wave and a full wave of spherical aberration, respectively. The response in the 
central plane is vastly superior to the response in the other two planes. Although the 
marginal curves are worse than the corresponding paraxial curves in the medium- 
frequency region, in the extreme regions of low and high spatial frequencies the con¬ 
verse is true. 



Spatial Frequency 

Fig. 15-17. Transfer function of circular aperture possessing a half-wave of 
spherical aberration in central, paraxial, and marginal receiving planes [15]. 



Fig. 15-18. Transfer function of circular aperture possessing one wave of 
spherical aberration in central, paraxial, and marginal receiving planes [15]. 







EFFECT OF ABERRATIONS ON TRANSFER FUNCTION 


633 


Parrent and Drane [16] have solved the slit-aperture problem for W 4 and W 2 using 
Simpson’s rule. Barakat [15] has utilized Gauss quadrature, and the curves shown in 
Fig. 15-19 and 15-20 were evaluated on a high-speed computer. The results are similar 
to those for the circular aperture, although the circular aperture curves are smoother. 

The extension of the analysis to include W 6 (fifth-order spherical-aberration coeffi¬ 
cient) is straightforward, and the aberration function is of the form 


W = W«p« + W 4 p 4 + W 2 p 2 
= Wdp« + «p 4 + /2p 2 ] 


(15-63) 


The addition of the two parameters a and (3 presents a bewildering array of possible 
combinations. A theory has been developed by Marechal which is based upon setting 
tolerances on the wave front in terms of the Strehl criterion. This theory is useful 
only for fairly small aberrations. 



Fig. 15-19. Transfer function of slit aperture possessing a half¬ 
wave of spherical aberration in the central, paraxial, and marginal 
receiving planes. 



Fig. 15-20. Transfer function of slit aperture possessing one wave 
of spherical aberration in the central, paraxial, and marginal 
receiving planes. 






634 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


15.3.1.3. Coma. Coma being an asymmetric aberration therefore has a transfer 
function with nonzero phase shift. Goodbody [17] has performed calculations and we 
quote from his work. The calculations are best performed when the aberration func¬ 
tion is expressed in rectangular coordinates. 

W = iW 3 i(x 2 -(- y 2 ) (x sin ip + y cos ip) (15-64) 

Two salient points emerge from Goodbody’s study: 

(а) The paraxial focus is the best focus in the sense that defocusing results in the 
deterioration of the transfer function in the low-frequency region. 

(б) The response for ip = 0 is better than for any other value of ip in the low-frequency 
region. 


Fig. 15-21. Transfer function for third- 
order coma of amount 0.63 A in two receiving 
planes. 




Fig. 15-22. Transfer function for third- 
order coma of amount 1.89 A in two receiving 
planes. 


The two points are illustrated in Fig. 15-21 and 15-22. The phase shift introduced by 
coma is nonlinear and is the cause of the harmonic distortion of the image. If the phase 
shift were linear with frequency, the effect would merely be a shift of the diffraction 
image; see [18]. 

15.3.2. Computation of Transfer Function of an Actual System. The fundamental 
problem of the transfer function from the point of lens designers is to find the functional 
relation between the transfer function and the design data. This problem has not been 
solved. However, in a given lens system, it may be necessary to obtain the transfer 
function directly from the ray-trace data. 







EFFECT OF ABERRATIONS ON TRANSFER FUNCTION 


635 


This problem has been attacked by Barakat [12] and Barakat and Morello [19] using 
Gauss quadrature theory. Barakat’s treatment is very complicated; see the actual 
papers for computational details and numerical results. 

The schematic presented in Fig. 15-23 illustrates the sequence of basic steps. Once 
the wave front has been obtained it is then necessary to fit a curve by some appropriate 
approximation scheme, to yield a polynomial or rotational expression for W. Once 
the wave front is known three basic quantitites can be computed: 

(1) Transfer function 

(2) Distribution of illuminance 

(3) Total illuminance 

The distribution of illuminance due to a point source can be computed by evaluating 
the KirchhofF diffraction integral Eq. (15-15). It is also of interest to know the total 
illumination in the various rings of the diffraction pattern, that is the fraction L of the 
total energy that falls within a circle of radius u 0 about the axial point in a given re¬ 
ceiving plane. Obviously L vanishes when Vo is zero and approaches unity as i» 0 becomes 
infinite. Although the illuminance isophotes (lines of constant illuminance) are 
extremely complicated, the corresponding isophotes of total illumination are smooth 



Fig. 15-23. Schematic of steps required for computation of transfer function and 
associated functions from design data. 







































636 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


functions. This is a consequence of the fact that the integration over the illuminance 
is a smoothing operation in the sense that the average properties of the diffraction pat¬ 
tern are brought to the foreground. 

Other information in this area can be found in [20] and [21]. 

15.4. Effect of Apodization on the Transfer Function 

The theory of apodization is concerned with the possibility of determining the ampli¬ 
tude distribution over the exit pupil (pupil function) in order to achieve some prespeci¬ 
fied distribution of illuminance over a designated receiving plane in the image field. 
The usual attempts involve an expansion of the amplitude distribution over the exit pu¬ 
pil (pupil function) in a convenient set of functions, e.g., Hermite, lambda, and Legendre. 

There are two theoretical approaches to apodization. The first and simplest is to 
choose some pupil function and to determine the resultant transfer function, spread 
function, etc. Second and more difficult is the synthesis problem; that is, given a 
prespecified quantity, determine the required pupil function. 

15.4.1. Altering High- or Low-Frequency Response by Apodization. Two im¬ 
portant general apodization theorems are: 

(1) If the pupil function weighs against the center of the aperture, then in the high- 
frequency region the transfer function increases over that of an Airy system. 

(2) If the pupil function weighs against the edge of the aperture, then in the low- 
frequency region the transfer function increases over that of an Airy system. 

As two examples which illustrate these theorems, consider a slit aperture with pupil 
the aperture; the latter against the edge of the aperture. The transfer functions for 
functions given by Eq. (15-27) and (15-28). The former weighs against the center of the 
aperture; the latter against the edge of the aperture. The transfer functions for these 
two pupil functions are shown in Fig. 15-24 and 15-25 for three values of a (= 0,0.5,1.0). 



Spatial Frequency 

Fig. 15-24. Transfer function of apodized slit aperture having 
pupil function Ai(/3). 

15.4.2. Luneberg Apodization Problems. Luneberg [22] formulated a series of 
important apodization problems, but gave the explicit solution to the first only. For a 
solution of the remaining problems see [23]. 

The most important of the theorems arises from the first Luneberg problem, which is 
to determine the pupil function that maximizes the central illuminance (essentially the 
Strehl criterion) subject to the condition that the total energy passing through the 



MERIT FACTORS 


637 



Fig. 15-25. Transfer function of apodized slit aperture having 
pupil function A 2 (l 3). 


aperture is constant. The final result is the pupil function which yields the maximum 
Strehl criterion, i.e., the Airy pupil function {A = 1). In other words, any apodization 
scheme will lower the Strehl criterion. 


15.5. Merit Factors 

15.5.1. Linfoot Quality Factors. The most important quality factors are the three 
proposed by Linfoot [24]. These factors determine on a statistical basis the degree to 
which an optical system reproduces various selected aspects of the object. 


15.5.1.1. Relative Structural Content. The first quality factor (and in some respects, 
the most important) is the relative structural content F defined in the spatial frequency 
domain by 


i r p 

F = — I |T(to x , a>„)| 2 0(co x , coy) d(Ojc d(Oy (15-65) 

t-'O J — oo J —oo 


where 



0((Oj, Wy) d(l)x d(j)y 


(15-66) 


and O(ot) x ,u)y) is the power spectrum of the object. In the special but important case of 
a flat object spectrum [0(a> x ,cej l/ ) = constant], F becomes 

F'=y j J \T((o x ,coy )\ 2 d(o x d(Oy (15-67) 

where F 0 is now the area of the spatial-frequency domain over which the transfer 
function exists. The contribution to F by the transfer function is always positive 
because T appears as a squared quantity. Thus the effects of spurious resolution are not 
accounted for using this quality factor. 

15.5.1.2. Fidelity Defect. The second quality factor is the fidelity defect defined 
in the frequency domain by 


i r°° r°° 

0 = — I I [1 — T((i) X , (Oy)] 2 0(a)x, (Dy) d(l)jr d(Dy (15-68) 

Oo J — oo J —oc 

The effect of spurious resolution is now taken into account because the transfer function 
now enters linearly (as well as quadratically) into 6. The fidelity-defect factor is a 







638 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 

measure of the degree to which the maxima and minima of the object and image are 
superposed. 

15.5.1.3. Correlation Quality. The correlation quality Q is given by 

1 r x r°° 

Q = — I T((0 X , (Oy)0((D X , 0)y) d(tJx (15-69) 

^0 J — oo J— oo 

= \(F + 0) 

As the second equation shows, Q is a linear combination of the first two Linfoot quality 
factors. In the case of an object with a flat spectrum, Q becomes 

i r r 

Q'=— I I T(w x , toy) d(o x du) u (15-70) 

C/0 J — oo J — oo 

which is the Strehl criterion introduced previously. Thus, Q is a generalization of 
the Strehl criterion. 

15.5.2. Plane of Best Focus. There is no general method for determining the plane 
of best focus because it depends upon the object being viewed. The investigations for 
three important cases are summarized: 

(1) For a point source 

(2) For a sine-wave target 

(3) For a random-detail target 

15.5.2.1. Point Source. The plane of best focus for a point source is that in which the 
Strehl criterion is a maximum. This definition gives a clear-cut answer only for 
small aberrations in systems with spherical aberration and defocusing. When the 
aberrations are large the answers are not unique and in fact the Strehl criterion is 
almost useless as a merit factor for determining the plane of best focus. The theory 
predicts that, for a system suffering third-order spherical aberration and defocusing, 
the plane of best focus is the central plane halfway between the paraxial and marginal 
planes. For values of W 4 > 2.5A, the Strehl-Richter theory fails. The Strehl-Richter 
theory is closely related to the Marechal theory previously mentioned. The Marechal 
theory does not deal with single aberrations, but tolerances are set on the mean square 
value of the wave front; it requires that the mean square deformation of the wave front 
with reference to a spherical wave front be minimized. 

15.5.2.2. Sine-Wave Target. The plane of best focus for periodic detail (sine-wave 
target) is that receiving plane for which the transfer function is a maximum at a speci¬ 
fied frequency; i.e., the plane of best focus depends upon the spatial frequency of the 
target. Figure 15-26 illustrates how the plane of best focus varies as a function of 
defocusing for a half wave of spherical aberration. Note the shift of the plane of best 
focus to the marginal focus (/x = 1) as the spatial frequencies are increased. A similar 
curve for one wave of third-order spherical aberration for both slit and circular apertures 
is shown in [16]. 

15.5.2.3. Random Detail. In practice one does not usually encounter point-source 
or sine-wave targets only but rather random distributions. This case differs from the 
two previous ones because statistical considerations enter. As the square of the 
transfer function is related to the mean square fluctuations in the image, a reasonable 
criterion as a measure of the plane of best focus is the relative structural content F 


MERIT FACTORS 


639 


<V 

C/3 

C 

o 

a 

c/3 

CD 

05 

£ 

£ 


a> 

c 

aJ 



Fig. 15-26. Plane of best focus for sine wave targets. 



Fig. 15-27. Relative structural content of 
circular aperture having spherical aberra¬ 
tion and defocusing. 


defined by Eq. (15-65). The plane of best focus in the presence of random detail is 
that plane in which F is a maximum. The plane of best focus will depend not only upon 
the system through the transfer function but also upon the object via its power spectrum. 
To illustrate the criterion, take the case of a flat object spectrum and a circular aperture 
suffering third-order spherical aberration. The results of the computations are shown 
in Fig. 15-27 for W 4 = 0.5A and 1.0X. The plane of best focus lies approximately in the 
central plane /x = 0.5. 




640 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 

15.6. Frequency Response Calculations from Lens Design Data* 

Frequency-response techniques have not yet reached a state of development such 
that they can be used for the synthesis of optical systems, although as a description of 
performance, and for some analysis steps, they are very useful. The Institute of Optics 
at the University of Rochester has a detailed design procedure programmed on a 1620 
IBM computer [25]. The method is based on the standard aberration polynomial, and 
includes a frequency-response calculation in the procedure. Some of Dr. Hopkins’ com¬ 
ments about frequency-response calculations are paraphrased: It is an exceedingly diffi¬ 
cult program to write. If design requirements are stringent, wave-front errors are 
reduced to a minimum. If not, there is a compromise at best and not much point in the 
time and expense of the frequency-response calculations. Frequency response is a 
function of wavelength and direction; what wavelengths and directions, how many, and 
what weighting they should have should be applied to any average. 

Frequency response can be calculated on the basis of physical optics. Miyamoto [6] 
discusses the calculations and their relationships in detail. Results in brief are: 
The region from 0 to 2\ waves of optical path difference is difficult to determine by 
approximate methods. As long as the geometrical image is inside the diffraction image, 
it is valid to use the geometrical response. 

15.6.1. Computer Calculations of Frequency Response. A recent article in 
Applied Optics describes the theory for performing some of these calculations. As an 
example, and for possible use or adaptation, the FORTRAN frequency-response program 
as used by the Institute is given below. 


FREQUENCY-RESPONSE PROGRAM FOR THE IBM 1620 

Abbreviations Used in Frequency-Response Program 

R = radius of a circular area at the image 
E = energy within the circle of radius R 
AR = r = mean of two adjacent R values 
DE = AE = difference of two adjacent E values 
N = number of R values = number of E values 
J = running integer 1 through N 
RSC = (sec 0) ,/2 

AN = 0 = obliquity angle of chief ray in image space 
EN = n = w/2tt = spatial frequency 
X = wr(RSC) = argument of Bessel function 
AJ = first order Bessel function, J 0 
T = modulation transfer factor 


^Contributed by Don Szeles, Institute of Science and Technology, The University of Michigan. 




FREQUENCY RESPONSE CALCULATIONS FROM LENS DESIGN DATA 641 


FLOW CHART-MODULATION TRANSFER FUNCTION 







































































642 OPTICAL FREQUENCY-RESPONSE TECHNIQUES 

FORTRAN STATEMENTS-MODULATION TRANSFER FUNCTION 


DIMENSION R(21),F(21),AR(20),DE(20) 

READ1,N 

1 FORMAT (12) 

D02,J=1,N 

2 READ3,E(J),R(J) 

3 FORMAT (E15.8.F15.8) 

RE AD4,RSC,AN 

4 FORMAT (E15.8,E15.8) 

NQ=N—1 

D05,I=1,NQ 

AR(I)=(R(I+l)+R(I))/2.0 

5 DE(I)=E(I+1)-E(I) 

6 READ7,EN 

7 FORMAT (E15.8) 

SUM=0.0 
D012,I=1 ,NQ 

X=6.2831853*EN* AR(I)*RSC 
IF(X-7.0)8,11, U 

8 Y=X*X/4.0 
W=Y 
EM=13.0 
SIG=—1.0 

9 Y=W*( Y/(EM*EM)+SIG) 

SIG=—SIG 

FM=EM—1.0 
IF (EM—1.0)10,10,9 

10 AJ=1.0—Y 
GO TO 12 

11 Y=1.0/(8.0*X) 

Z=Y* Y 

P=1.0-Z*(4.5-459.375*Z) 

Q=-Y*( 1.0-Z*(37.5-7741.875*Z)) 

SQ=SQRT (3.1415927*X) 

A=(P—Q)/SQ 
B=(P+Q)/SQ 

AJ=A*SIN(X)+B*COS(X) 

12 SUM=DF(I)*AJ+SUM 
T=SUM/F(N) 

PRINT13,EN,T,AN 

13 FORMAT (5HEN = ,E13.8,5H T = ,E13.8, 7H AN = ,E13.8) 
D015,J=1,N 

15 PRINT16,J,E(J),J,R(J) 

16 FORMAT (2HE(,12,4H) = ,E13.8,6H R(,12,4H) = ,E13.8) 
GO TO 6 

END 


45 CARDS 


OPTICAL RESPONSE MEASURING EQUIPMENT 
INPUT TO 1620 FREQUENCY RESPONSE 


643 



FIELD 1 

FIELD 1 

FIELD 2 

FIELD 2 

FIELD 3 

FIELD 3 

CARD NO. 

SIZE & TYPE 

VARIABLE 

SIZE & TYPE 

VARIABLE 

SIZE & TYPE 

VARIABLE 

1 

2, I* 

N 





2 thru N+l 

15, E* 

E(J) 

15, E 

R(J) 



N + 2 

15, E 

RSC 

15, E 

AN 



N+3.N+4,... 

15, E 

EN 






Type I is an integer (e.g., 1, 12, 20). 

Type E is a signed decimal number followed by a signed power of ten for relocating the decimal point 
(e.g., —.842 E —3 —» .000842). A maximum of 8 places after the decimal point is allowed and the 
power of ten exponent must be > — 99 and < +99. 

*A11 numbers must be right justified in the field. 


15.7. Optical Response Measuring Equipment 

There is no standard way for measuring optical response. The techniques usually 
involve the preparation of a series of spatial sine waves and a method for measuring 
the image contrast. Two designs appear in the recent literature [26,27]; others can 
be developed. The essence of one device is quoted from [27]: 

A stabilized tungsten ribbon filament lamp serves as a source, the filtered light 
passing through a condenser system uniformly illuminates a fine slit. The slit then 
serves as a self-luminous incoherent object. The lens under test forms an image of the 
slit in the front focal plane of a microscope objective. The microscope objective then 
presents the image to sinusoidally varying masks of fixed spatial frequency and ampli¬ 
tude which are mounted in frequency pairs on a rotating drum. The image transmitted 
by the sinusoids is integrated by a 931-A RCA photomultiplier tube. The diffuser in 
front of the phototube uniformly illuminates the photocathode elements. The output 
of the photomultiplier tube is then presented to a potentiometer pen recorder. 

The area masks are arranged in frequency pairs, where one area mask is stepped 90° 
out of phase with the other. In this way ten spatial frequency pairs are arranged, all 
the peaks of the cosine area masks are in line with one another on the drum, then 
automatically the area masks representing the sine component would be in line. Thus 
when a slit image is presented to the area masks and its transmission measured and 
recorded, the peaks on the chart recording would then represent the real and imaginary 
parts of the transfer function for each spatial frequency available on the drum. 

For an image which shows no phase shift effect, all the peaks on the chart recording 
corresponding to the transmission of the sinusoids (imaginary part) will be equal to 
each other and equal to the half amplitude zero spatial frequency area mask which in 
turn is equal to the average transmission of the area masks. These peaks represent 
the baseline for the measurement, and is the self-normalization constant for the meas¬ 
urement. The half-amplitude normalization target has an area transmittance equal 
to the average chart transmittance but is a zero frequency target. Thus at some 
frequency, when the image suffers a shift in phase a peak appears below the normaliza¬ 
tion curve; this phenomenon, most common when the system suffers from defect of focus, 
is called "spurious resolution.” A curve connecting all the maxima of the cosines is 
then the real part of the transfer function of the system relative to the half-amplitude or 
average transmittance. A curve connecting all the sine terms is a plot of the imaginary 
part vs. frequency. 

The drum carries ten frequency pairs, ranging from 0.201 lines/mm for the lowest 
to 8.20 lines/mm for the highest frequency. The apparatus is equipped with three 



644 


OPTICAL FREQUENCY-RESPONSE TECHNIQUES 


microscope objectives which magnify the frequencies such that it is possible to obtain 
a range from 1.79 lines/mm to 730 lines/mm. The drum rotates at two speeds. At 
the high speed the transfer function can be seen on an oscilloscope where rapid adjust¬ 
ments can be made for proper position and normalization. When the drum is switched 
to low speed, which is approximately 1 rpm, a permanent record can be made on the 
recorder. Thus it can be seen that after proper normalization, it takes only minutes 
to record directly the transfer function of an optical system. For an experienced 
operator a complete characteristic can be measured for a lens system for at least twenty 
positions through focus in one-half hour. 

References 

1. E. H. Linfoot and E. Wolf, Proc. Phys. Soc., 66B, 145 (1953). 

2. T. Asakura and R. Barakat, Japan J. Phys., 30, 728. 

3. R. Barakat, Progress in Optics, 1 North Holland Publishing Co., Amsterdam, 67-108. (1961) 

Chap. 3. 

4. E. L. O’Neill, J. Opt. Soc. Am., 46, 285 (1956). 

5. W. Steel, Rev. optique, 32, 4(1953). 

6. K. Miyamoto, Progress in Optics, 1 North Holland Publishing Co., Amsterdam, 33-66. (1961) 

Chap. 2. 

7. L. J. Cutrona, E. N. Leith and L. J. Porcello, Trans. IRE., AC-4, 137 (1959). 

8. E. L. O’Neill, Trans. IRE., IT-2, 56 (1956). 

9. M. Bom and E. Wolf, Principles of Optics Pergamon, New York, (1959). 

10. H. H. Hopkins, Wave Theory of Aberrations Clarendon Press, Oxford, (1950). 

11. A. Marechal and M. Francon, Diffraction, Structure des Images (Editions de la Revue d’ Optique, 
Paris (1960). 

12. R. Barakat, J. Opt. Soc. Am., 52, 985 (1962). 

13. H. H. Hopkins, Proc. Roy. Soc., 231, 91 (1955). 

14. G. Black and E. H. Linfoot, Proc. Roy. Soc., 239A 522 (1957). 

15. R. Barakat, J. Opt. Soc. Am., 53, 324 (1963). 

16. G. Parrent and C. Drane, Optica Acta, 3, 195 (1956). 

17. A. M. Goodbody, Proc. Phys. Soc., 72, 411 (1958). 

18. M. De and B. K. Nath, Optik, 15, 739 (1958). 

19. R. Barakat and M. Morello, J. Opt. Soc. Am., 52, 1328A (1962). 

20. R. Barakat, J. Opt. Soc. Am., 51, 152 (1961). 

21. E. Wolf, Phys. Soc. (London), Repts. Progr. in Phys., 14, 95 (1951). 

22. R. K. Luneberg, Mathematical Theory of Optics, Brown University Providence (1944). 

23. R. Barakat, J. Opt. Soc. Am., 52, 276 (1962). 

24. E. H. Linfoot, J. Opt. Soc. Am., 46, 740 (1957). 

25. Image Evaluation Techniques, III, "Summer Course Notes,” Institute of Optics, College of 
Engineering and Applied Science, The University of Rochester, Rochester, N.Y. (1963). 

26. R. R. Shannon and A. H. Newman, Appl. Opt., 2, 365 (1963). 

27. T. Tsurata, Appl. Opt., 2, 371 (1963). 

Bibliography 

Barakat, R., J. Opt. Soc. Am., 52, 264 (1962). 

Conrady, A. E., Applied Optics and Optical Design, 1 Dover (1957). 

Duffieux, P., LTntegrale de Fourier et ses Applications a (Optique, Besancon, Faculte des Sciences, 
(1946). 

Felgett, P., and E. H. Linfoot, Trans. Roy. Soc., 247, 369 (1954). 

Marechal, A., Rev. optique, 27, 73 (1948). 

O’Neill, E. L., "Selected Topics in Optics and Communication Theory,” Boston University Physical 
Research Laboratory (1958). 

Picht, J., Optische Abbildung Braunschweig Vieweg, (1931). 

Steward, G. C., The Symmetrical Optical System, Cambridge University Press, (1928). 


Chapter 16 

SPATIAL FREQUENCY 
FILTERING 

James Alward 

The University of Michigan 


CONTENTS 


16.1. Introduction. 646 

16.2. Basic Mathematical Relationships. 646 

16.2.1. Fourier Transform. 646 

16.2.2. Properties of the Two-Dimensional Fourier Transform. 646 

16.2.3. Two-Dimensional Fourier Transforms in Polar Coordinates . . . 647 

16.2.4. Dirac Delta Function. 648 

16.2.5. Products and Convolutions. 648 

16.2.6. Autocorrelation Functions and Wiener Spectra. 649 

16.3. Analysis of Spatial Frequency Filtering. 649 

16.3.1. Scanning Aperture Space Filters. 650 

16.3.2. Fixed-Field Moving-Reticle Space Filters. 650 

16.3.3. Scanning-Field Moving-Reticle Space Filters. 652 

16.3.4. Circular Sectored Reticles. 653 

16.4. Fourier Transforms of Common Space Filters. 654 

16.4.1. Rectangular Aperture. 654 

16.4.2. Circular Aperture. 654 

16.4.3. Infinite Parallel-Spoke Reticle. 654 

16.4.4. Infinite Checkerboard Reticle. 654 

16.4.5. Parallel-Spoke Reticle Limited by Rectangular Aperture .... 654 

16.4.6. Parallel-Spoke Reticle Limited by Circular Aperture. 655 

16.4.7. Checkerboard Reticle Limited by Rectangular Aperture .... 655 

16.4.8. Circular Sectored Reticles. 655 


645 























16. Spatial Frequency Filtering 


16.1. Introduction 

In this chapter, the basic mathematical relations employed in spatial frequency 
filtering are summarized, the usual approaches to spatial filter analysis are presented, 
and the commonly encountered space filter expressions are tabulated. Almost no de¬ 
rivations are given here. Reference [1] contains a reasonably thorough, intuitive 
introduction to most of the concepts involved in spatial filtering. 

16.2. Basic Mathematical Relationships 

16.2.1. Fourier Transform. The Fourier transform and inverse transform for a 
two-dimensional spatial pattern s(x,y ) are: 


S(k x , k y ) = f [ six, y) e j 2 Mk x x+k v y) dx dy 

J — 00 J — 00 

(16-1) 

six,y)=r r Sik x ,k y )e i 2 irik * x+k v y) dk x dk y 

J — 00 J — X 

(16-2) 


Spatial frequencies in the x and y directions are represented by k T and k y , respectively. 
The arbitrary spatial pattern s(x,y) may be considered to be real for all incoherent 
infrared systems. S(k x ,k y ) is in general complex. |S(& x ,& y )| is its amplitude term; 
e x ’ y is its phase term. Fourier transform pairs are denoted by a double headed 
arrow: s(x,y) <-*■ Sik x ,k y ). The condition for the existence of Eq. (16-1) and (16-2) is 
that the following inequality holds: 

I I \s(x,y)\ 2 dxdy < (16-3) 

j — 00 J — 00 

16.2.2. Properties of the Two-Dimensional Fourier Transform. Some useful 
properties of the two-dimensional Fourier transform, which may be easily derived from 
the definition, are given in this section. 

16.2.2.1. Space Scaling. If s(:t,y) and S(k x ,k y ) are Fourier transform pairs, then the 
following transform relationship exists: 

s(ax ' 6y) ~r^ s (M) (l6 - 4) 

16.2.2.2. Space Shifting. If six,y) is shifted by a constant in each direction, its 
amplitude spectrum is unchanged, but its phase spectrum in each direction is modified 
by a term linear with space frequency. 

six -x 0 ,y- y 0 ) S(k x , k y ) e ~ l2Mk * x » +k v y » ) (16-5) 

16.2.2.3. Space-Frequency Shifting. The corresponding transform pair for a shift 
in space frequency is: 


** S(k t - k y - *„) (16-6) 


646 



BASIC MATHEMATICAL RELATIONSHIPS 


647 


16.2.2.4. Space and Space-Frequency Differentiation. 


d m d n 


dx m dy n 


s(x,y) (j27rk x ) m (J2nky) n S{k x , k y ) 


(16-7) 


d m d n 

(-j2nx) m (-j2iTy) n s(x,y) ++ S{k x , k v ) 


(16-8) 


16.2.2.5. Conjugate Functions. If s(x,y) is a complex spatial function, then for its 
conjugate s*(x,y), the following transform pair holds: 


8*(X,y) S*(~ k X ,~ ky) 


(16-9) 


if s(x,y) is real, then 


s*(x,y) = s(x,y) <-> S(k x , ky) 

S(k x ,ky) = S*(— kx,~ ky) 
S*(k X , ky) = S(— k x> — ky) 


(16-10) 


(16-11) 


(16-12) 


16.2.2.6. Symmetrical Spatial Functions. If s(jc,y) is both real and symmetrical 
about the origin; i.e., if s(jc,y) = s(— x — y), then S(k x ,k y ) is real and symmetrical, and 

r oo r oo 

S(k x> k y ) = 4 s(x,y) cos 2rr(k x x *+■ k y y) dx dy (16-13) 

Jo Jo 

r oo /*oo 

s(x,y) = 4 S(k x , k y ) cos 2rr{k x x + k y y) dk x dk y (16-14) 

Jo Jo 

16.2.2.7. ParsevaVs Theorem. The two-dimensional expression for Parseval’s 
theorem for two real spatial functions Si(x,y) and s 2 (jc,y) is 

f f si(x, y)s 2 (x, y) dx dy = f j Si(—k x , — k y )S 2 (k x , k y ) dk x dk y (16-15) 

J — 30 J — QC ► — 00 J — 00 


-r r 

J — oo J — 00 


S l(k x , ky)S 2 ( — k X , —ky) dk X dky (16-16) 


= r r Sx*{k X , ky)S 2 {k X ,ky) dkxdky (16-17) 

J — 00 J — » 

= r P Sl(k X> ky)S 2 *(k Xf ky)dk X dky (16-18) 

J — 00 J — 00 


16.2.3. Two-Dimensional Fourier Transforms in Polar Coordinates. The rela¬ 
tions between rectangular coordinates (x,y) and ( k x ,k y ) and polar coordinates (p,0) and 
(k p , if/) are: 

x = p cos 6 k x = k p cos if/ 

y = p sin 0 k y — k p sin ip 

x 2 + y2 = p2 k x 2 + k y 2 = kf 

dxdy = p d p dO dk x dk y = k p dk p dip ^ 


(16-19) 








648 


SPATIAL FREQUENCY FILTERING 


The Fourier transform has the form 

/•oo f2n 

C l , \ I ( a \ — j2 it k p(cos 6 cos if + sin 6 sin if) , , 

S(k p ,if/)= s(p,0) e p pdOdp 

Jo Jo 

r<x> cltr 

-[ I s( P ,»)e- i2 ’ k "‘ ,co,te -* > pded P (16-20) 

Jo Jo 

The inverse Fourier transform has the form 

roc r2.1T 

s(p, 6) — I I S(k p , if/) e l2lTkpP cos (e * ] kp dty dk p (16-21) 

Jo Jo 


If the spatial function is not dependent on 6, then the Fourier transform pair fors(p) 
takes the form 

S(k p ) = f s(p)J 0 (kpp)p dp (16-22) 

J o 

s(p) = [ S(k p )Jo(kpp)kp dkp (16-23) 

Jo 

where J 0 is the Bessel function of the first kind and order zero. 

16.2.4. Dirac Delta Function. The spatial Dirac delta function 8(x,y) is used to 
represent a finite energy source or a finite transmittance concentrated into an arbi¬ 
trarily small region of the plane. For example, 5(x— x 0 ,y— yo) is used to represent a 
"point” source at (x 0 , yo). The energy density at (x 0 ,yo) is infinitely large. 

Mathematically, the delta function can be defined in terms of its sifting property: 



8U—Xo, y—yo)s ( x, y) dx dy = s(x 0 , yo) 


(16-24) 


where s(x,y) is an arbitrary function continuous at (x 0 ,yo). In terms of this definition, 
the Fourier transform of 8(x—x 0 ,y— yo) is easily found to be e j2n(k * x o + k v y o ) thus 
establishing a Fourier transform pair for the Dirac delta function. 

A delta function in terms of only one spatial variable 8(x — x 0 ) may be interpreted 
physically as a line source or a line transmittance of infinite length and arbitrarily 
small width, along the line x = x 0 . 

Another property of the delta function is: 


8(ax,by) 


1 

\a\\b\ 


8(*,y) 


(16-25) 


16.2.5. Products and Convolutions. If the inverse Fourier transform of the 
product of two space-frequency functions Si(k x ,k y ) and S 2 (k x ,k,,) is taken, the result is 
the convolution of the inverse Fourier transforms of the two functions: 

roo /*x 

s(x,y) = I Si(£, T7)s 2 (x-£, y—q) d^ drj (16-26) 

J —oo J —oo 

Similarly, if the Fourier transform of the product of two spatial functions Si(x,y) and 
s 2 (x,y) is taken, the result is the convolution of the Fourier transforms of the two func¬ 
tions: 



ANALYSIS OF SPATIAL FREQUENCY FILTERING 649 

S(k x ,ky)=f I Si(k ( , k v )S 2 (k x -k(, ky-kv) dk( dkr, (16-27) 

J—oo J — oo 

16.2.6. Autocorrelation Functions and Wiener Spectra. The autocorrelation 
function w(g ,r}) is defined as a joint mean of a random process [ 1 ]. 


w(£,r))=[ I SiS 2 p 2 (si, i; S 2 , 17 ) ds t ds 2 (16-28) 

J—00 J— oc 

where Si and s 2 are two sample functions evaluated at two randomly chosen points, 
Ui,yi) and {x 2 ,y 2 ) respectively, and p 2 is the second-order joint probability density 
function of the random process which generated Si and s 2 . £ and p are equal to X\ — x 2 

and y x — y 2 , the displacement coordinates between (xi,yi) and (x 2 ,y 2 ), respectively. The 
random process is assumed to be stationary in terms of second-order statistics. If 
the random process is also ergodic, the autocorrelation function may be represented 
in terms of an average over space of a single sample function of the random process: 

1 f A f B 

w(£, 17 ) = lim —— s(x,y)s(x+£,y+i 7 ) dx dy (16-29) 

» 4 AB J-A J-B 

B —► oo 

The condition for the existence of Eq. (16-28) and (16-29) is that total mean square 
average of the random function be bounded; i.e., that 

TTd f f |s(x,y )| 2 dx dy < » (16-30) 

^->00 4 AB J_ A J_ B 

which is a less stringent condition than expression (16-3). An autocorrelation function 
for functions satisfying (16-3) may be represented as follows: 


w(%, 17) = 



s(x, y)s(x+£, y+17) dx dy 


(16-31) 


The Wiener spectrum, W(k x ,k y ), the analogy of the power spectrum in electrical 
systems analysis, is defined simply as the Fourier transform of the autocorrelation 
function: 

W(k x ky) = r f*w(£, 7j) e - j2ir(k ** +k » r,) d{ dr) (16-32) 

J —00 J —00 

If the spatial pattern s(x,y) satisfies expression (16-3) the Wiener spectrum may be 
represented directly in terms of S(k x ,k y ): 

W(k x ,ky ) = |S(£.r,Ml 2 (16-33) 

Since the Fourier transform of a random (continuing) function does not exist [1], 
the Wiener spectrum must be used whenever a space frequency representation is needed. 

16.3. Analysis of Spatial Frequency Filtering 

The basic expression for the time-varying output v(t) of a spatio-temporal filter is 


v{t) = 



r(x, y, t)s(x, y) dx dy 


(16-34) 


where r(x,y,t ) is a general expression for a spatio-temporal filter, and s(x,y) is an 
arbitrary input scene. The infinite limits are for generality; actually r{x,y,t ) is zero 




650 


SPATIAL FREQUENCY FILTERING 


outside a finite region. This expression applies to scanning-aperture space filters, to 
fixed-field moving-reticle space filters, and to scanning-field moving-reticle space filters. 


16.3.1. Scanning-Aperture Space Filters. For a scanning aperture moving with a 
velocity u x parallel to the x axis, r(x,y,t ) is represented as r(x— u x t,y) and the output 
is given by 


v(t) = 



r(x—u x t, y)s(x, y) dx dy 


(16-35) 


The Fourier transform V(f) (also called the output spectrum) of this expression is 
given by 

V( r>=~ J* «*(£. k)s(£, k,)dk y de-set 


If the input scene is represented only in terms of its Wiener spectrum W(k x ,k y ), the 
output power spectrum is 



(16-37) 


If there is also a velocity component u y in the y direction, the expressions for V(f ) 
and <£(/*) are 


V(f) = 


<*>(/*) = 



(16-38) 

(16-39) 


16.3.2. Fixed-Field Moving-Reticle Space Filters. One type of fixed-field moving- 
reticle space filter is simply an infinite reticle pattern r x (x—u x t,y ) scanning over the 
limited input scene s(x,y)r a (x,y), where r a (x,y) is the fixed field of view of the space 
filter. 

The Fourier transform of an infinite parallel-spoke square-wave reticle, assuming the 
spokes are parallel to the y axis, is an array of delta functions (along the k x axis) whose 
magnitudes are determined from the Fourier series of a square wave: 


Roo(k X , ky) = ^ 8(k X> ky) + V 8{k X ~ [2 n - l]ko,ky) 

z _ \Zn — l )tt 


n = 1 


+ S To—""TT 8 (£x+ [2 n — l]ko,ky) ( 16 - 40 ) 
— (2n — 1)7 t 

n = l 

where ko is the fundamental space frequency, the period being the width of one spoke 
pair. The output spectrum V(f) from scanning the endless reticle over the limited 
scene with a velocity u x in the x direction is: 


V(f) w j j S(£x, f d^i 

*{ H-) + s -i)*-l 

I 2 \lixJ n = 1 (2^1 1)7T lUx J 


+ 2 


(- D"- 1 

(2n - 1 )tt 


8 


— + (2 n — 1)^ 
u x 



(16-41) 












ANALYSIS OF SPATIAL FREQUENCY FILTERING 


651 


The output is periodic, hence the expression is simply the transform of a Fourier series 
whose coefficients are determined from the square wave, but are further modified 
by the spectrum of the limited scene. If the input scene is represented only in the 
form of a Wiener spectrum W(k x ,k y ), the output power spectrum <£(/*) is 



W(U,Cy) 



(if 




s [f - (2 n - l)*o] 
[(2n - 1 )t r] 2 


+ 


00 

I 


n=1 


+ (2 n - 1)* 0 ] 
[(2 n — 1)7r] 2 


(16-42) 


The Fourier transform of an infinite checkerboard square-wave reticle, assuming that 
the pattern is oriented so the square edges are parallel to the coordinate axes, is a 
double array of delta functions along the 45° lines which bisect the right angles formed 
by the coordinates in the ( k x ,k y ) plane. The magnitudes are determined from the 
Fourier series of a symmetrical triangle wave [1]. 


Roo(k X , ky) — ~ 8(k X , ky) + 


A [( 2 / 1 - l)ir] 


8[k x —(2n—l)k 0 , k y — [2n—l ] k 0 


+ 2 


.-e, [(2n — l)ir] 2 


+ 2 


e, [(2 n - l)irp 


8 [ k x +{2n— 1)£ 0 , k y +{2n— 1)/j 0 ] 


8[k x —(2n—l)k 0 , k y +(2n— l)&o] 


+ S T7o-— V 2 8[k x +(2n— l)k 0 , k y -(2n-l)k 0 ] 

L(2n — 1)7 tJ 2 


(16-43) 


where k 0 is the fundamental space frequency in either the x or y directions, the period 
being twice the width of one checkerboard square. The output spectrum V(f) from 
scanning the endless checkerboard with a velocity of u x in the x direction over a scene 
limited by r a ( x,y) is: 

v <» -i i s (£) i: t: - «■) ^ * • 

+— 1 n 1, 4- - (2n - i)*»i r r s({„{,) 

u x £ [(2 n - 1)7r] 2 L u x J J_ x 


1 “ 

+-2 


u x " [(2 n 


n=l 


*[£ + “"-■>*•] r.r. si< -" 

Ra\-£—U- (2n-l)*.-£,]</{x di„ 











652 


SPATIAL FREQUENCY FILTERING 


+ u * [(2n - 


i__ 8 [X_ (2n _ 1A ]££ S(uw 

2n-l)4o-{ y ] <*£* d£„ 


+ i 1 s [£ +<2n -«*•] £ £ s «~ w 

R„M-—{,, (2n-l)* 0 -&,] di, di, (16-44) 

If the input is described only in terms of its Wiener spectrum, the output power spec¬ 
trum <I> (f) corresponding to Eq. (16-44) is 

$(/•) =XI 8 (X) J* J" <#*<#, 

+ z i i£ - (2n - i)fc> £ £ w 


n=l 


|fia[^-U(2n-l)i.-C„]| ! <<£*<«» 


+ u * »?i [(2n - 


I_ 8 rx 

- l)ir]< [u. 


+ (2 n — l)k 0 


iff 

J •'—00 •'—0 

(2»- l)t.-{»]| 2 d{x dl. 


l,) 


1 » 
+ — Y 


^ r * s[--(2 n ~i)d r r wuz.i.) 

u x ", [(2n - 1 )tt-] 4 Ux J 

|R.[^-Cr,-(2n-l)A 0 -£„]| 2 dl x dty 


1 ® 


Mx ", [(2n - 1 )tt] 


X_ 6 [X +(2 „_ mo ]££^, c 


IrJX- 

Lu x 


£x, (2n-l)ko-iy 


dC,X d^y 


(16-45) 


16.3.3. Scanning-Field Moving-Reticle Space Filters. The scanning-field moving- 
reticle space filter has both a moving reticle and a scanning aperture; however, the 
velocities of the two elements are not necessarily equal nor even in the same direction. 
In the previous case of the reticle scanning a scene limited by a fixed aperture, the 
time-varying output was periodic. The effect of having the aperture move is one of 
modulating the periodic signal and thus spreading the signal energy into frequency 
bands about the original signal harmonics [2]. Two scanning situations are considered 
below. 

The first situation concerns both the reticle r(x,y) and the aperture field stop a{x,y) 
moving in the x direction. The velocity of the reticle with respect to the scene is u xr , 










ANALYSIS OF SPATIAL FREQUENCY FILTERING 


653 


the velocity of the aperture with respect to the scene is u xa , and the velocity of the 
reticle with respect to the aperture is u xra , where u Tra = u xr - u xa . Usually u xr is 
greater than u xa . The time-varying output is given by 

r oo roc 

u(t) = s(x, y) a(x — u xa t, y)r(x — u xr t, y) dx dy (16-46) 

J— oo J — 00 

The corresponding output spectrum V(f) is 

V( ^~u^l x \_ x J_ x S(kx > k v) A *( Uxra ,kv ~ kv ) R *^~uT a > k 'v) dk * dk y dk 'v 

(16-47) 

The second situation is concerned with the reticle scanning in the y direction with a 
velocity of u yr with respect to the scene, and the aperture moving in the x direction 
with a velocity of u xa with respect to the scene. Also, the reticle does not move with 
respect to the aperture in the x direction, so that the reticle also has a velocity of u xa 
in the x direction with respect to the scene. The time-varying output is given by 


v(t) 


roc r oo 

= s(x,y)a(x—u X at,y)rix—u xa t,y—u V rt)clxdy (16-48) 

J— oo J — oc 


The corresponding output spectrum is 


v</)=—r r r s^, f ) 

Uyr J — oc J — co J—oo \ Uyr ) 

R*(k' x , 1 — dk.dk„ dk, 

\ Uyr J 


(16-49) 


Expressions (16-47) and (16-49) involve a double convolution in the k y and the k x 
directions, respectively. 

For inputs represented only in terms of their Wiener spectra, the output power spec¬ 
trum expressions corresponding to Eq. (16-47) and (16-49) are, respectively: 


<k/)=—P r r wik^k,) A( kruir f ,k v -k\\ 

U X ra J—o o J —oo J-oo \ U X ra / 


R ( f kxUxa k')i\ dkxdkfjd k' y 

V U X a / I 

<Hn —- r r r w<k„k„) al.-h x ,k,+ kiUia ~ f \ 

Uyr J — oo J —oo J— oo V Uyr / 


(16-50) 



f kj-Ujrg^ 


x> 


U 


yr 


d k x 1 dky dkjc 


(16-51) 


16.3.4. Circular Sectored Reticles. The circular sectored reticle, which is also 
known as the "wagon-wheel” reticle or the episcotister, is shown in Section 16.4. This 
reticle is best represented in polar coordinates as a periodic variation in the 6 direction. 
The Fourier transform R{k p ,\\t) of a circular sectored reticle with n black-white spoke 
pairs and with a radius of a is given by 
















654 


SPATIAL FREQUENCY FILTERING 


aJ i (27 rak p ) 

^-1- 


71 K P p=l 


(—i')(2p-l)n ri 

2 sin ap ~ 1)n ' 1 ’ J. 


2nak• 


zJ(2p-l)n(z) dz 

(16-52) 


J( 2 p-i)n(.z) denotes the Bessel function of the first kind and order (2 p — l)n. R(k p , if/) 
is a complex function when n is odd, and is real when n is even. If the original reticle 
function is rotated with an angular velocity cot so that r(p,0) becomes rip, 6—cot), then 
R(k p ,\lt ) becomes R(k p ,4i+cot). In other words, the Fourier transform rotates with the 
same velocity in the opposite direction. 

16.4. Fourier Transforms of Common Space Filters 

Ten pairs of the more common space filters and their Fourier transforms are illus- 
strated and summarized below for use in the mathematical expressions of Section 16.3, 
especially Section 16.3.2. Solid black in the figures indicates a transmission of 0; 
white indicates a transmission of 1. 

16.4.1. Rectangular Aperture. Figure 16- 1(a) show a simple rectangular aperture 
with dimensions a x 6. The Fourier transform of this aperture is 


„ , r , N sin 7 raft* sin nbk y 

ky) 2 L L 


(16-53) 


One quadrant of \R a (k x , k y )\ is shown in Fig. 16-1(5). 

16.4.2. Circular Aperture. Figure 16-2(a) shows a circular aperture of radius a. 
The Fourier transform of this aperture is 


Ra ( kx, ky) 


adxdl'na V&x 2 + k y 2 ) 
Vkx 2 + ky 2 


(16-54) 


One quadrant of \R a (k x ,ky)\ is shown in Fig. 16-2(6). 

16.4.3. Infinite Parallel-Spoke Reticle. A parallel-spoke reticle which is infinite 
in extent in both directions is shown in Fig. 16-3(a). Its Fourier transform is given 
by Eq. (16-40) and is shown in Fig. 16-3(6). 

16.4.4. Infinite Checkerboard Reticle. A checkerboard reticle which is infinite in 
extent in both directions is shown in Fig. 16-4(a). Its Fourier transform is given 
by Eq. 16-43 and is shown in Fig. 16-4(6). 

16.4.5. Parallel-Spoke Reticle Limited by Rectangular Aperture. A parallel- 
spoke reticle limited by a rectangular aperture of dimensions a x 6 is shown in Fig. 
16-5(a). The Fourier transform of this reticle is given by either of the following expres¬ 
sions. 


R{ k X y ky) ^ 


1 sin 7ra&.r sin nbk 


nkj 






n=1 


(— l)"- 1 sin 7ra[^ x — (2 n — l)/g 0 ] sin rrbky 
(2n—l)77- 77- [6j. — (2n — l)£o] nky 



(— l)" -1 sin Tra[k x 4- (2/i — 1 )&q] sin Trbk u 
(2n — l)7r Tr[k x + (2n — 1)& 0 ] rrky 


(16-55) 


R{kx, ky) 


sin nbky sin TT{xJ2)kx sin nakx 
nky nk x sin nx 0 k x 


(16-56) 


Figure 16-5(6) shows \R(k x ,ky)\ for k x > 0. 




















FOURIER TRANSFORMS OF COMMON SPACE FILTERS 


655 


16.4.6. Parallel-Spoke Reticle Limited by Circular Aperture. A parallel-spoke 
reticle limited by a circular aperture of radius a is shown in Fig. 16-6(a). The Fourier 
transform of this reticle is 


R ( k X , ky) = — 
2 


Ji(27raV^j- 2 + k y 2 ) 

\4* 2 + k y 2 


+2 


(- l)"" 1 a e/i(27raV [k x ~ (2 n - l)&o] 2 + &„ 2 

(2n - 1)t r V[A,-(2n- 1)A 0 ] 2 + V 


+ 


I 


(- l)"-^ J,(277-aV[^(2n - 1)& 0 ] 2 -t- k y 2 

(2n - Dtt V[*, + (2n- m 0 ] 2 +A tf 2 


(16-57) 


One quadrant of \R(k x> k y )\ is shown in Fig. 16-6(6). 

16.4.7. Checkerboard Reticle Limited by Rectangular Aperture. A checkerboard 
reticle limited by a rectangular aperture with dimensions a X 6 is shown in Fig. 16-7(a). 
The Fourier transform of this reticle is given by either of the following two expressions: 


d/ l n_l sin TTCikx sin nbky 
{ x ’ y) ~ 2 nkx 


+ 2 


1 _ sin 77 a[k x — (2 n — l)fe 0 ] sin 776 [k y — (2 n — 1)& 0 ] 

^ [(2/i - D 77] 2 77[^ tf — (2n — 1)& 0 ] 77^^ — (2n — 1)& 0 ] 


+ 2 


1 _ sin na[kx 4- (2n — 1)& 0 ] sin 7rb[k y + (2 n — 1)& 0 ] 

[(2n — l)n] 2 Tr[k x +(2n—l)ko] n[k y + (2n — l)k 0 ] 


+ 2 


1 _ sin 7ra[k x —2n— 1)&q] sin 776 [k y -f (2n — 1 )& 0 ] 

£ x [(2n — D77] 2 Tr[k x —(2n — l)k 0 ] Tr[k y + {2n — T)ko\ 


+ 2 


1 


sin TTa[k x + (2 n — l)/e 0 ] sin 776 [k f) — (2 n — 1)/j 0 ] 


~[[(2n — D77] 2 7r[k x + (2n — 1 )& 0 ] rr[k y — {2n — \)ko) 

(16-58) 

. . _ sin Tr{xol2)k x sin ir{xJ2)ky sin rrakx sin rrbky x Q 

R(kx , A„) = 2- 7 -r-:- t — - r cos 77 77 + ky) 

nkx rrk y sin 77X 0 «x sm 77X 0 « y 2 (16 59) 


\R(kx,k y )\ is shown in Fig. 16-7(6) for k x > 0. 

16.4.8. Circular Sectored Reticles. Circular sectored reticles with one, two, and 
four spoke pairs are shown in Figs. 16-8(a), 16-9(a), and 16-10(a). The Fourier trans¬ 
form of a circular sectored reticle with n spoke pairs is given by Eq. (16-52), which 
consists of a term independent of n and a summation of terms dependent on n. The 
term which is independent of n is simply one-half of expression (16-54), which is shown 
in Fig. 16-2(6). Figures 16-8(6), 16-9(6), and 16-10(6) show only the summation terms 
of R{k p ,ilf) after the first term has been removed. 

References 

1. IRIA State-of-the-Art Report on Spatial Frequency Filtering, Report No. 2389-87-T. The Uni¬ 
versity of Michigan, Ann Arbor, Michigan (To be published). 

2. V. J. Ashby, The Principles of Space Filtering in the Image Plane of a Simple Optical System, 
Space Technology Laboratories, Canoga Park, Calif. (1961). 































656 


SPATIAL FREQUENCY FILTERING 



Fig. 16-2(6) 




















SPATIAL FREQUENCY FILTERING 


657 



Fig. 16-3(6) 



Fig. 16-3(a) 



Fig. 16-4(6) 
























658 


SPATIAL FREQUENCY FILTERING 



Fig. 16-6(6) 



































SPATIAL FREQUENCY FILTERING 


659 



Fig. 16-8(6) 




















660 


SPATIAL FREQUENCY FILTERING 



Fig. 16-9(6) 



R(k , W 


Fig. 16-10(a) 



Fig. 16-10(6) 















Chapter 17 

CONTROL SYSTEMS 


K. R. Morris 

The University of Michigan 


CONTENTS 


17.1. Linear Systems. 662 

17.1.1. Basic Definitions. 662 

17.1.2. Determination of Transfer Functions. 664 

17.1.3. Methods of Analyzing Linear Systems. 668 

17.1.4. System Types and Performance. 685 

17.2. Sampled-Data Systems. 695 

17.2.1. Basic Definitions. 695 

17.2.2. Determination of Transfer Functions. 700 

17.2.3. Methods of Analyzing Sampled-Data Systems. 701 

17.2.4. Types of Sampled-Data Systems. 703 

17.3. Nonlinear Systems. 703 

17.3.1. Basic Definitions. 703 

17.3.2. Methods of Analyzing Nonlinear Systems. 705 

17.3.3. Specific Solutions. 712 

17.4. Design Methods. 718 

17.4.1. Gain Adjustment. 718 

17.4.2. Cascade or Series Compensation. 718 

17.4.3. Root-Locus Method. 722 

17.4.4. Optimum Transient Response Behavior 

for Torque-Saturated Systems. 724 

17.4.5. Miscellaneous Comments on Design and Compensation .... 724 

17.4.6. Statistical Design. 725 


661 
























17. Control Systems 


17.1. Linear Systems 
17.1.1. Basic Definitions 

Basic System. The basic system to be considered is shown in Fig. 17-1. Most 
systems, however complex, can be reduced to this basic form by applying the rules 
and techniques given in Sec. 17.1.2. 


R(s) 


C(s) 


r(t) 


c(t) 


Fig. 17-1. Basic feedback control system. 

In general, throughout this chapter, functions represented by lower case letters are 
functions of the time variable t; capital letters are used to represent the Laplace trans¬ 
form of the lower-case letter function, and are functions of the complex variable s = 
cr +joj. 

Typical Test Signals 

(1) The Delta function, 8{t — to), u 0 (t — t 0 ), is zero for all time except to, when its 

height is infinite and its area is unity; it is approximated by a short rectangular 
pulse of unit area. 

(2) The unit step, hit — t 0 ), U\ (t— to), is zero for all t < to and one for all t > to’, it is 

usually specified to be one-half at t = to. It is approximated by fast-rising 
voltage (relay or switch with mercury wetted contacts, semiconductor switch, 
etc.). 

(3) The unit ramp, u)(t — to)t, u> (t — to), is zero for all t < to, increasing with unity 

slope for t > to. 

(4) The unit parabola, a(t — to)t 2 , u 3 (t — to), is zero for all t < to, increasing at t 2 

for t > to. 

Time Descriptions of System Behavior 

(1) The time constant is the time it takes the system to reach (1/e) A, when the input 

is Ah(t) (see Fig. 17-2). 

(2) Rise time, t r , is the time it takes the system to move from 0.1A to 0.9A, when the 

input is Ah(t ) (see Fig. 17-2). 

(3) Time delay, t (i , is the time interval between the time of application of input, 

A/iU),tothetime the system reaches 0.5A (see Fig. 17-2). 

(4) Overshoot is (A,, — A)/A, with input Ah(t) and a peak value of output A,,. 

Overshoot is measured as a fractional or absolute value, or as a percent (see 
Fig. 17-2). 



662 









LINEAR SYSTEMS 


663 



(5) Time of response, or setting time, t s , is the time required for the system output 
(with input Ah(t )) to reach and stay within a certain tolerance (e.g., 5%) 
of the final value (see Fig. 17-2). 


( 6 ) 

(7) 

( 8 ) 


Time to peak, t p , is the interval between application of input Ah(t) and peak 
output. 

The position or displacement error is given by e = r — c (r and c defined as in 
Fig. 17-1). 


The velocity or rate error is given by e — r — c. = —, where x is any variable.^ 


(9) The acceleration or parabolic error is given by e — r — c. 



c/ 2 x\ 

!?)' 


Weighting Function. The weighting function g(t ) is the response of a system to a 
unit impulse input. 

Transfer Function. Transfer functions may be defined as follows. 


(1) The Laplace transform of weighting function 

G(s) = L {gU)} 

where L{g(t)} = Laplace transform of g(t) = G(s). 

(2) The ratio of Laplace transforms of input and output 

r = L{c(t)} = C(s) 

L{r(t)} fits) 

(3) The frequency response function when it exists (i.e., for stable systems) [1]. 






























664 CONTROL SYSTEMS 

The basic forms of transfer functions are: 


(a) Gain or sensitivity G(s) = K 

( b ) Differentiator G(s) = ts 

(c) Integrator G(s) = (ts) -1 

(d) First-order lead G(s) = 1 + ts 

( e ) First-order lag G(s) = (1 + ts) -1 

(/) Second-order lead G(s) = s 2 + 2£a>„s + co„ 2 

( g ) Second-order lag G(s) = (s 2 + 2£co„s -I- a) n 2 ) -1 

(/i) Time delay G(s) = 

Stability 

Absolute Stability. If the output of a linear system is bounded for any bounded input, 
the system is said to be absolutely stable (we exclude neutrally stable systems since 
in practice they do not exist). 

Relative Stability. Relative stability is the change necessary to make a system 
unstable. 

Minimum Phase System. A minimum phase system is one whose characteristic 
equation has roots with only negative real parts, hence the minimum phase shift for 
a given amplitude characteristic. 

Gain Margin. Gain margin is the additional gain necessary to make the system 
unstable. 

Phase Margin. Phase margin is the additional phase shift necessary to make the 
system unstable. 

Crossover Frequency. The crossover frequency o) c , or the frequency to which the 
loop is closed, is the frequency at which system gain is 0 db. 

17.1.2. Determination of Transfer Functions. Transfer functions can be deter¬ 
mined from signal flow graphs, block diagrams, Bode plots, and Laplace transforms 
of transient signals. 

17.1.2.1. Signal Flow Diagrams [2] 

Introduction. Signal flow diagrams are directly applicable to circuits (electrical, 
mechanical, etc.) and furnish a method for obtaining the transfer function. They 
can also be applied to block diagrams for simplification. 

The circuit equations are 


5) CLijXi = 0 

i = 0 


7 = 1,2, 3,.. 


The equations can be rewritten 


x j ^ tijXi j — 1, 2, 3,... 

t' = 0 


tij — 


aij 

a.ij - 1 


when j # i 
when j — i 



LINEAR SYSTEMS 


665 


Figure 17-3 shows the generalized diagram if each x, is a node, and if n is 3. The 
transfer function of the entire circuit can be found by applying signal flow rules and 
diagram simplification. 




Fig. 17-3. General signal flow 
diagram for a 3-terminal network. 


Signal Flow Rules 

(а) Signals travel only in the direction of arrows. 

(б) A signal is multiplied by the transmittance of the branch Uj. 

(c) The value of the signal at any node is the sum of the signals entering the node. 

( d ) The signal value of any node is sent along all branches leaving the node. 

Rules for Simplifying Signal Flow Diagrams 

(а) Parallel paths can be combined by summing transmittances (Fig. 17-4). 

(б) Series paths can be combined by multiplying transmittances (Fig. 17-5). 




Fig. 17-4. Example of parallel 
path combination. 


Fig. 17-5. Example of series 
path combination. 


(c) Terminal and source ends of branches can be moved. To move the terminal 
end of a branch £ 3 , (see Fig. 17-6), a new node W) must be defined, at which all incom¬ 
ing branches (except the one to be moved) terminate. e\ is coupled to the old node 
(e i) by transmittance 1. A branch from the original node (ei) to any node (c 5 ) must 








666 


CONTROL SYSTEMS 



Fig. 17-6. Example of procedure for moving 
the terminal end of a branch. 


have its source moved to e/ and must receive a branch from the source node (e 3 ) of the 
moved branch; this new branch will have transmittance ti 5 tsi. The terminal end of 
the branch is moved from e\ to e 2 , multiplying its transmittance by that of the path 
along which it moved (i.e., ^ 31 ^ 12 )- 

To remove the source end of a branch from e 3 (Fig. 17-7), each source which had a 
branch terminating at e 3 must now have a branch terminating at e 2 . For example, 
for e A (see Fig. 17-7) the transmittance is £ 43 ^ 2 , etc. 

( d ) Self loops can be eliminated by multiplying the incoming branch transmittances 
by 1/(1 - t kk ) (Fig. 17-8). 











LINEAR SYSTEMS 


667 


t 13 t 32 




Fig. 17-7. Example of procedure for removing source end of a branch. 




Fig. 17-8. Example of the elinination of a self loop. 


17.1.2.2. Block Diagrams [3]. Basic rules for the simplification of block diagrams 
are illustrated in Fig. 17-9, where the contents of each box is the transfer function 
of that box. 


17.1.2.3. Bode Diagrams. Experimental curves are obtained which relate system 
attenuation and phase to frequency (i.e., frequency response curves). Bode’s method 
(see 17.1.3.5) is used to draw approximate asymptotes to the attenuation curve; the 
phase plot can be used to determine "break” points. The approximate curve can be 
corrected by adjusting £ in quadratic terms (see 17.1.3.1 and 17.1.3.5). Further cor¬ 
rections, such as Linville’s method [2], are available. 


17.1.2.4. Transient Response Analysis 

General Method. Apply a transient e(t) and record the open-loop system output 
c(t) (some useful inputs are found in Sec. 17.1.1). Then the ratio of the Laplace trans¬ 
forms of response and input, G{s ) = C(s)/E(s), yields the transfer function G(s). For 
complicated c(t), one can use approximations made from a series of simple functions 
(such as steps, triangles, impulses, rectangles, etc.), and then find the transform of 
each; the total transform is the sum of the transforms of the individual components. 

Guillemin’s Impulse Method [2]. An impulse is applied to the system to obtain 
g(t); g(t) is then approximated by a sequence of straight lines, impulses, parabolic 
curves, cubic curves, or other higher powers of t, yielding g*(t). g*(t ) is differentiated 
until only impulse functions remain; then 




2 

k = 1 


a k e 


~jut k 


(»" 









668 


CONTROL SYSTEMS 


(1) 




( 6 ) 


a 


G(s) 


b 


a 

G(s) 







a 

1 



CKs) 



Fig. 17-9. Basic rules for block diagram simplification. 


where G*(ja )) is the approximate transform ofgU) 
a k is the amplitude of the Mh impulse 
tk is the time of the kth impulse 

v is the number of derivatives taken to obtain impulses 
n is the total number of impulses used. 

Approximating g(t) by straight lines is equivalent to approximating g{t) by para¬ 
bolic segments. 

17.1.3. Methods of Analyzing Linear Systems. A linear system can be analyzed 
by considering the differential equations describing it, by its weighting function, 
by Nyquist methods, by the Bode method, by log modulus plots, and by root locus 
methods. 

































































LINEAR SYSTEMS 


669 


(7) 


( 8 ) 





(9) 


b -» f \ 


( 10 ) 


<*■> — 


(ID 



( 12 ) 




Fig. 17-9 ( Continued ). Basic rules for block diagram simplification. 


17.1.3.1. Differential Equation Method 

Representation of the System. Sum the torques (forces, voltages, etc.) on the out¬ 
put and write the equation of motion (where p = dldt ), 

C(p ) = R(p) 

The characteristic equation is C(p) = 0. 

Absolute Stability Analysis. If the characteristic equation has any derivatives 
missing (lower than the order of the equation), or if the coefficients of the character¬ 
istic equation are not all of the same sign, the system is unstable. If the roots of the 
characteristic equation have any positive real parts, the system is unstable. A useful 
test for positive real roots is Routh’s Stability Test I [4]: 

Write the characteristic equation 

n 

C(s) — ^ a,s" _< = 0 
1 = 0 










































































670 


CONTROL SYSTEMS 


Form an array from the coefficients, as follows, continuing until the last row has only 
one entry: 

Go Go 

Oi 03 

b i 63 

Ci C3 


where 


6, 


— det 


ao 

Gi 


a 2 

g 3 


63 = — det 


Go 


ai 


G 4 

a 5 


Ci = — det 


a t 

6, 


a 3 
63 


The number of zeros in the right-half s-plane is indicated by the number of sign 
changes in the first column. Work can be simplified by dividing each element of a row 
by the absolute value of the first element of that row. If an entry in the first column 
is 0 , replace the zero by e and continue. 

If all elements of a row are 0, there are imaginary roots [5]. An auxiliary polynomial 
is formed from the last non-zero row; the order is n (the order of the original equation 
is m, the number of the last non-zero row is r, and n= m — r+ 1 ). The powers of s con¬ 
tained in this polynomial will either be all even or all odd. Differentiate and proceed. 

If the characteristic equation is written in terms of K (a gain constant), limits on K 
for stability can be determined such that the sign requirement is satisfied. 

Example 

(s — 2) (s + 3) (s ■+■ 5) (s 2 ■+■ 9) 
s 5 + 6 s 4 + 8 s 3 -I- 24s 2 — 9s — 270 


Row 


1: 

2 : 

3: 

4: 

5: 


1 

8 

-9 

1 

4 

-45 

[ 6 ] 

[24] 

[-270] 

1 

9 


[41 

[36] 


-1 

-9 


[-5] 

[-45] 


-2 

[01 

0 

form 


one root with positive real part, one pair of complex, conjugate roots. 








LINEAR SYSTEMS 


671 


An alternate form is Routh’s Test II: 
Form the array: 


CL\ a 0 0 


O 3 0-2 CL\ 


0 0 
0 0 


a 5 a 4 


a :i a 2 a 1 


For stability, all determinants formed by the elements in the upper left corner must 
have the same sign, i.e., 


a 1 
a 3 


a 0 
a> 


a, 

Oo 

0 

as 

a 2 

ai 

a 5 

a 4 

a 3 


Steady-State Behavior c s (t). The system equation can be written as 


c(t ) = r(t) 


2 ai p { 


i= 0 


_pA'^ b m p r 


1=0 


Form the power series 

c{t ) = r(t) 

Example 


p~ N 1 - 7 —^— p~ . 
' b () — a 0 


then 


c(t) = -— r(t) = (1 - rp+ r 2 p 2 - . . .)r(« 

rp 1 


if r(t) = 8(t), c s (t ) = 0 
if r(£) = Ah(t), c g (t ) = A 
if r (0 = c s (£) = cot — tco 

Alternately, for r(t ) = Ah(t), set all derivatives in the characteristic equation to zero. 
Example 


(p 2 + 4p + 253 )c(t) = r(t)(p 3 + 32) 


c s (t) 


32 A 
253 


Transient Behavior cAt). Assume a solution of the characteristic equation 

n 

cAt ) = 2 

i=i 

where the A, are the eigenvalues of the characteristic equation and the C, are deter¬ 
mined from the initial conditions. 














672 


CONTROL SYSTEMS 


For many systems the performance is dominated by two complex poles (no zeros), 
such that 

G(s) ~ (s 2 ■+■ 2£co„s + OJ n 2 ) -1 
Then the system has the following characteristics: 

(а) £ is the damping factor 

(б) con is the undamped natural frequency 

(c) a>„(l — £ 2 V 2 = cod, the damped frequency 

id) time to peak t p = n/cod 

(e) settling time (to ±2%)t g ~ 4/£w„ 
if) overshoot = exp (— C,co n t P ) 

(g) rise time (10 to 90 %)t r ~ 2.2/£co„ 

( h) logarithmic decrement 8 

8 = 1/nln Xo/XnT = 2 t™£/(1 - £ 2 W 2 

where x 0 is the amplitude of the first overshoot (measured from the final value) and 
x„t is the amplitude of the signal after n cycles of oscillation. £ is the damping ratio 
of a second-order system. This is a valid measure only for lightly damped systems 

(£ < 0 . 1 ). 

17.1.3.2. Weighting Function Method 

Representation of the System. The weighting function git) is the output obtained 
when a unit impulse is applied to the input. Step inputs, ramp inputs, etc., can be 
used if the output is differentiated (once for step inputs, twice for ramp inputs, etc.). 

Absolute Stability Analysis. The James-Weiss stability criterion [11 states that 
for absolute stability 

f |g(r)| dr < * 

Jo 

Time Behavior from Weighting Function. Determine the output of the system 
either graphically or analytically by using 

c(t) = [ e(t) git — r) dr 
Jo 

This method is not as useful for determining damping factors, frequency response, 
etc.; but it does give c(t ) directly. 

17.1.3.3. Nyquist Method [61 

Representation of the System. Write the open-loop transfer function as 

KGis) = KA(s)/B(s) 

On Fig. 17-10 locate all poles and zeros (singular points) on the s-plane which lie on 
the jco axis. Select a clockwise contour enclosing all poles and zeros in the right-half 
plane (a semicircle at infinity is chosen). This contour should bypass each pole and 
zero on the jco axis by a small counter-clockwise semicircle. 

Evaluate KG(J(o) = \KG(Joj)\ jjb for enough values of co that KGijoo) can be plotted 
as a function of <f> (the Nyquist or polar plot in the KG plane). The plot is symmetrical 
around the real axis. As co travels around the semicircle enclosing the singular points 
on the jco axis, the Nyquist plot contour moves through an angle of approximately 
±180n° at very large radius, (n = order of singularity; -I- for poles, — for zeros.) Often, 
Gijco) is plotted instead of KGijco) (G-plane Nyquist plot) (see Fig. 17-11). 




LINEAR SYSTEMS 


673 



Fig. 17-10. s-plane contour for 


G(s) = 


_ 1 _ 

sis + 3 )(s 2 + 4)' 


Imaginary 

joe 



-jx 


Fig. 17-11. G-plane contour (Nyquist plot) for 

_ 1 

s(s + 3)(s 2 -f 4) 


Real 


G(s) = 












674 


CONTROL SYSTEMS 


Absolute Stability Analysis. First, determine the number of clockwise encircle¬ 
ments (AO of the critical point. 

(a) In the KG plane, the open-loop critical point is the origin. 


( 6 ) 


In the KG plane, the closed-loop critical point is —1, since 


G c (s) 


KG(s) 

1 + KG(s ) 


(c) In the G plane, the closed-loop critical point is — UK 

Encirclement can be determined by tracing around the KG (or G) contour in a clock¬ 
wise direction; if the critical point is on the right, it is inside the contour, hence encircled. 
Next, determine the number of poles in the right-half plane (zeros of B(s)). Then 


Z = P — N 


where P = the number of right-half-plane poles circled by the s-plane 

N = the number of encirclements of critical points 

Z = the number of right-half-plane zeros in the system 

Steady-State Behavior. The steady-state behavior is determined by the manner 
in which the G or KG contour behaves as a>—»0. 


(а) If lim G = closed plot, the system had displacement error 

to —»o 

(б) If lim G = —j°°, the system has velocity error 

to —* 0 

(c) If lim G = — oo, the system has acceleration error 

<o —» 0 

( d ) If lim G = +j<*>, the system has rate-of-acceleration error 

to —* 0 

(e) Lower-order errors are 0 (see also Sec. 17.1.4). 

Transient Behavior. M circles (see Fig. 17-12) are the locus of constant ratio of 
open-loop to closed-loop magnitude. The coordinates of the center are 


-1 M 2 
KM 2 - 1 


y = 0 


The radius is given by 


1 M 
KM 2 - 1 


(а) M p is the peak amplitude of the closed-loop response, and G c (s) is given by the 
M circle tangent to the contour (Fig. 17-13). 

(б) cop, the frequency of resonance, is the frequency on the contour at the point of 
tangency with the M p circle (Fig. 17-13). 

(c) The magnitude of gain at any point of the contour, ois determined by first 
drawing an M circle through the point of interest. Then a ip line is drawn 
through the origin tangent to the M circle. If i// is the angle made by the i fj 
line and the negative real axis, then (see Fig. 17-13) 

1/M = sin ifj 

( d ) The gain constant K corresponding to any point of the contour is determined by 
a line perpendicular to the real axis and passing through the point of tangency 
of the M circle and the ifj line (see Fig. 17-13). 

(e) Frequencies for which the locus is to the right of the line M = 1 (circle of infinite 
radius) are attenuated. 






LINEAR SYSTEMS 


675 



The best transient response (best because the transfer function exhibits the least 
peaking) is given by the K value derived from \pmax (the largest«// which is still tangent 
to the locus). 

Since the distance and angle from the origin to the locus give G(ja>) while that 
from the critical point — UK gives +1 IK + G{jw) (Fig. 17-14), the total system response 
G c (j(o ) is given by 


G c (s ) 


KG(s) 

1 + KG(s ) 


hence the ratio of the two distances gives the magnitude of G c (s), and the difference 
between the angles gives the phase shift. 

The phase margin is given by the angle formed by the vector of length 1 IK which 
intersects the G contour, and the negative real axis. N circles (Fig. 17-15) are circles 
























676 


CONTROL SYSTEMS 


M = 1 Imaginary 



Fig. 17-13. Example of transient behavior 
determination from M-circles. 



Fig. 17-14. System gain from Af-circle. 















LINEAR SYSTEMS 


677 


Imaginary 



Real 


o 

os 

1 

o 

Center 
x = - 1/2 

y 

Radius 

±10° 

±2.84 

2.88 

±20° 

±1.37 

1.46 

±30° 

±0.866 

1.00 

±45° 

±0.5 

0.707 

±60° 

±0.29 

0.577 

±70° 

±0.18 

0.531 

±90° 

0 

0.500 


Fig. 17-15. N-circle loci for polar (Nyquist) plots. 


of constant phase difference between input and output signals; they are characterized 
by a center 

x = —1/2, y = +1/2N 
N = tan id,, — Oc) 


where 0 o is the open-loop phase shift 
6c is the closed-loop phase shift 
The radius of the N circles is given by 



At the frequency at which a locus crosses an N circle, the difference between input 
and output phase is given by the value of tan -1 N. 

17.1.3.4. Inverse Nyquist Method. When components His) are found in the feed¬ 
back loop such that 

KGis ) 

Gds) = 


1 + KG(s)H{s) 



















678 


CONTROL SYSTEMS 


it is convenient to consider 

[G r (s)] _1 = + AF(s)H(s) 

K Lt{s) 

The sum can be performed graphically by vector summation at each o>. M~ x circles 
are concentric about — 1/K; other characteristics carry over from the Nyquist method 
in a similar manner. 

17.1.3.5. Bode Method (log Magnitude and Phase vs log Frequency). 

Representation of the System. Write the transfer function in terms of basic 
forms, i.e., 

Q/ g \ _ _ K(TiS + 1) (T2S + 1) . . . _ 

S A (?«S 4- 1) (jftS + 1) (s 2 + 2£o ) n s + (On 2 ) . . . 


Next, construct a straight-line approximation to the Bode diagram by summing these 
basic components (Fig. 17-16 and 17-17). The diagram can be corrected by applying 
the curves of Fig. 17-18 and 17-19. An example is shown in Fig. 17-20. If the system 
is non-minimum phase, the phase portrait will be modified as shown in Fig. 17-186. 



Fig. 17-16. Straight-line approximations to basic forms 
for Bode plot, |G(jo>)|. 











PHASE — - 20 log 


LINEAR SYSTEMS 


679 



s 


3 


s 


1/s 

1 /s 2 

1/S 3 


Fig. 17-17. Straight-line approximations to basic forms 
for Bode plot, / G(j(o) = <f>. 


1 l_ 1 1.31 2 



Fig. 17-18. Corrections for Bode plot of first-order basic forms. 


Phase rs - 1 

FOR NONMINIMUM PHASE SYSTEMS 































PHASE OF g(j w ) 


680 


CONTROL SYSTEMS 



log < j ) 
(a) 



(b) 


Fig. 17-19. Corrections for Bode plot of second-order basic forms, 
(a) Amplitude correction. (b) Phase correction. 












LINEAR SYSTEMS 


681 



Fig. 17-20. Example of Bode plot for 


G(s ) = 


167(s + 60) 
sis + 10) 


(a) Amplitude of Gijco). ib) Phase of G(j a>). 


Absolute Stability Analysis. If the system gain is above unity (0 db) when the 
phase angle exceeds 180°, the system is unstable. 

Steady-State Behavior. The steady-state behavior is determined by the slope of 
the diagram: 

(a) If lim |G| = 0 slope, the system has displacement error 

<o —» 0 

(b) If lim | G| = —6 db/octave slope, the system has velocity error 

w —» 0 

(c) If lim |G| = -12 db/octave slope, the system has acceleration error 

(u -* o 

See Sec. 17.1.4. 

Transient Behavior. In general, the greater the phase margin, the smaller the 
transient (see Fig. 17-21). A totally equalized system is stable for all gain with a slope 
of -6 db/octave and a phase shift of-90° for 0 < w < oj c ; the response is almost purely 
exponential. A partially equalized system is stable for some gains, i.e., the slope is 
-6 db/octave for id near w c ; in this case, the response is somewhat oscillatory. Some¬ 
times a system is referred to by the slopes of the approximate response, e.g., 12-6-12. 






























PHASE ANGLE 20 1og|G(ja)l 


682 


CONTROL SYSTEMS 



Fig. 17-21. Example of Bode plot for G(s) = ■■■ ^ ^ 1 showing phase and gain margins. 

s*(s* + lOUs + lUu 


17.1.3.6. Log Modulus Method (log Magnitude vs Phase) 

Representation of the System. Write the transfer function as 

KG(s) = ^^- = |G(s)| / <t>(s) 

Plot 20 log | G(s) | vs <t>(s)-, the total system plot is the sum of the angle and magnitude 
curves for the system components (Fig. 17-22). 

Absolute Stability Analysis. If the system response diagram intercepts the 
±180° axis above 0 db, the system is unstable. 

Steady-State Behavior. Steady-state behavior is determined by the way the con¬ 
tour behaves as cj —*■ 0. See Sec. 17.1.4. 

(а) If <£—► 0°, the system has displacement error 

oj —* 0 

(б) If —90°, the system has velocity error 

to —* 0 

(c) If <£—> —180°, the system has acceleration error 

(o -* 0 

Transient Behavior. M and N contours (Nichols chart) (Fig. 17-23) can be used as 
in the Nyquist method; the phase margin is indicated as the contour intercept with the 
0-db axis, and the gain margin is indicated as the contour intercept with the 180° axis. 













LINEAR SYSTEMS 


683 


Phase 


, kfafg 10 ' PHASE MARGIN 

0 20 40 60 80 100 120 140 160 180 



Fig. 17-22. Example of log modulus plot for 

C ( 63.1(5+10) 

S s 2 (s 2 + 100s + 10 4 ) 


17.1.3.7. Root Locus Method 

Representation of the System. A root-locus plot shows the locus of closed-loop 
poles as the open-loop gain is changed. Consider the transfer function of the open-loop 
system G(s) = p(s)/q(s). The zeros of q(s) (i.e., poles of G(s))are marked on a diagram 
as x’s, the zeros of p(s ) as o’s. The rules for construction are as follows: 

(а) The loci start at poles. 

(б) The loci terminate at zeros. 

(c) The number of loci is the number of poles or zeros of G(s). 

( d ) If the coefficients of p and q are real, complex parts of the loci appear in con¬ 
jugate pairs. 

(e) The loci approach °° along asymptote lines at ± 180°/n, ± 5407n, ± 9007rc, 
etc., where n is determined by letting s get very large so that only the highest- 
order terms of p and q remain of importance. Then 


G(s) - K/s* 







684 


CONTROL SYSTEMS 



PHASE ANGLE OF G(ju>) 

Fig. 17-23 M and N contours on log modulus plot (Nichols chart). 


( f ) The point at which these asymptote lines meet is determined by writing 


Then 


G(s) = K 


s m + ais m_1 + a 2 s m ~ 2 . . . . a m 
s m+n + 6iS m+n_1 + . . . . am + n 


Si 


b\ ai 


n 


or 


2 poles — 1 zeros 

(number of finite poles) — (number of finite zeros) 

( g) Real-axis loci: Those portions of the real axis to the left of an odd number of 
critical frequencies (poles + zeros) are part of the root loci. 

( h) Imaginary axis intersections: Apply the Routh test to p(s) + q(s ) (see Sec. 
17.1.3.1). 


(i) Angles of departure and arrival: At any point on the locus the sum of angles 
of vectors from poles must be 180° + 360n°. The angle of departure can be 
chosen to satisfy this requirement. 

O') The point-a at which loci leave the real axis: Choose a point Si just off the axis 
(e). The transition from —a to Si must yield no net change of angles from poles 
and zeros { 6 1 + 0 2 + d 3 = 0). Replace angles by their tangents and solve for a. 
If Si is above the real axis, conjugate complex roots will cause a net decrease 
in angle equal to 


2 eyl((3 2 + y 2 ) 

















LINEAR SYSTEMS 


685 


where y is the distance along the real axis from the complex poles (to the right 
of—a) to —a 

/3 is the distance of the complex root from the real axis 

( k ) Sum of loci: If p(s) 4- q(s ) is of degree n and the coefficient of s n is 1, the coeffi¬ 
cient of s" -1 is the negative of the sum of the zeros. 

(/) Product of loci: If p(s)/q(s ) contains a pole at the origin, the constant term of 
p(s) 4- q(s ) is directly proportional to K. Choose a point z on the locus. The 
gain at z is given by the value of K which makes [p(s) -f q(s)]/(s + z) = 0. K at 
any point can also be determined by multiplying the lengths of the vectors from 
the open-loop poles and zeros. 

Absolute Stability Analysis. The system becomes unstable for values of K such 
that the roots are in the right-half plane. 

Steady-State Behavior. To determine steady-state behavior, determine the num¬ 
ber of poles at the origin of the open-loop system. (See also Sec. 17.1.4.) If the number 
of poles is 

(а) 0, the system has displacement error 

(б) 1, the system has velocity error 

(c) 2, the system has acceleration error 


Transient Behavior. The contribution of a pole to system behavior is Ae (<r+jajU . 
The more negative cr, the less the contribution compared with roots closer to the joj axis. 
The frequency response function can be found as follows: 

(а) Choose an w on the ju) axis 

(б) Let G(s ) = KKsKjio — r x )(j(i) — r 2 ) • • • where jio — n is the distance from the 
chosen jio to the ith root (for a given K). 

(c) Graphical procedures are found in [2]. 

The time response can be found directly as follows: 


(a) Write the open-loop transfer function as 


KG(s) = 


K(aiS -f bi)(a 2 s -f b 2 ) * • • 
s m (cis + di)(c 2 s 4- d 2 ) • • • 


KK s (s + 1/ti)(s + 1/t 2 ) 
s m (s 4- l/r a )(s + 1 hb) • 


where 


K s = K,K 2 K 2 


tit 2 t 3 


TaTftTc 

(6) Take the inverse Laplace transform of 

KK,R(s ) 


C(s) = 


(s — r a )(s — r b ) • • • 
where rj is the location of the ith root in the closed-loop system. 

(c) Graphical procedures are found in [2). 

17.1.4. System Types and Performance 

17.1.4.1. Introduction. Consider the system presented earlier in Fig. 17-17, where 

C(s) 


KG(s ) 


E(s) 


, the open-loop transfer function 







686 


CONTROL SYSTEMS 


G c (s) 


C(s) _ KG(s) 
R(s ) 1 4- AG(s) 


the closed-loop transfer function 


E(s) 

R(s) 


1 

1 -f AG(s)’ 


the error-response function 


A system is characterized by the open-loop transfer function as: 


AG(s ) 


C(s ) _ K(st\ 4- l)(sr 2 +!)••• 
E(s) s n (st„ 4- 1)(st/> 4- 1) • • • 


The system is classified 

(а) Type 0 if N = 0 

(б) Type 1 if AT = 1 
(c) Type 2 if N = 2 

If a Bode plot is made for C(s) for a given A(s), the plot may be considered an approxi¬ 
mation to an inverse time plot, i.e., t ** 1/w. 

Error coefficients are defined classically as follows: 

(а) Position error coefficient: Kp = lim G(s) 

S — 0 

(б) Velocity error coefficient: K r — lim sG(s ) 

s —» 0 

(c) Acceleration error coefficient: K„ — lim s 2 G(s) 

S —» 0 

Error coefficients can also be found from 

E(s) 1 1 + _s_ + _^ 

R(s) 1 + KG(s ) 1 + Ap A r A a 

Steady-state errors can be found for various inputs, since 


(а) if r(t) = h(t) y e ss = d + A p ) _1 

(б) if r(t) = t, e ss = (A,.) -1 


(c) if r(t) = t 2 , e ss = (A a ) _1 

This is useful only for systems for which the poles of G(s) are in the left-half s-plane. 

17.1.4.2. Type 0 System (Simple Speed Controllers, Feedback Amplifiers). The 
open-loop transfer function of a Type 0 system is given by (see Fig. 17-24a) 


AG(s) = Kp 


1 + TS 

The closed-loop transfer function of a Type 0 system is given by 


P __ ix p 

c ts(, 1+K p ) 1 +K P 

if r << (1 +K„). 

The error (see Fig. 17-24 6) is given by 


E(s ) _ 1 + TS ~ 1 

R(s) ts + (1 4- Kp) 1 4- K p 

















LINEAR SYSTEMS 


687 


w 

£ 


3 3 

O W 


bd 

O 


o 

CM 



-45 


-90 


o 


o 


(a) 



0 ° 


-45 


o 


-90 


o 


(b) 

Fig. 17-24. Type 0 system Bode plot, 
(a) Open loop. ( b ) Closed loop. 


The response of the Type 0 system to a step input is described by the following equa¬ 
tions (see Fig. 17-25). 

A 

r(t) = Ah(t) R(s ) = — 

s 


C(s) = 

E(s ) = 


AK P 

s{ 1 + K p ) 
A 

s( 1 + Kp ) 


c(t) ~ 


e(t) = 


Kp 

1 + K P 
A 

(1 + Kp) 


A Type 0 system develops a displacement error in response to a displacement input. 
The response of the Type 0 system to ramp input is described by the following equa¬ 
tions (see Fig. 17-26). 


r(t) = Wit 


R(t) 


oh_ 

s 2 


C(s) 


Kp CO i 

1 + Kp s 2 


at) = 


p CO j t 

1 + Kp 





































688 


CONTROL SYSTEMS 


CO 

z 

2 


Al 

AK 

_E_ 

K + 1 
P 


_A 

K 


P 


_ 

+ 1 L 

0 


t 


r(t) 

c(t) 


e(t) 


Fig. 17-25. Response of Type 0 system 
to step input. 




Fig. 17-26. Response of Type 0 
system to ramp input. 


Fig. 17-27. Response of Type 0 
system to parabolic input. 


E(s) = 


1 (O, 

1 + K p s 2 


e(t) = 


a >it 

1 + K P 


A Type 0 system develops a velocity error in response to velocity input. 

The response of the Type 0 system to parabolic input is described by the following 
equations (see Fig. 17-27). 


r(t ) = ait 2 


,.v «' 

c(t) = — 

oO 


C{s) = 

Kp Of i 

c(t) = 

1 + Kp s 3 

E(s) = 

1 a 

II 

1 +K d s 3 


K p at 2 
2(1 + K p ) 

at 2 


2(1 + K p ) 

A Type 0 system develops an acceleration error in response to an 

17.1.4.3. Type 1 System (Classical Motor-Driven Loops). The 
function of a Type 1 system is given by 


acceleration input, 
open-loop transfer 


C(s) K v 

E(S) S(1 + T m s) 


or, if T m < < 1 (see Fig. 17-28a), 


c(s ) K v 


E(s) s 






















LINEAR SYSTEMS 


689 





Fig. 17-28. Type 1 system Bode plot, (a) Open loop. 
(6) Closed loop, (c) Error response. 


The closed-loop transfer function of a Type 1 system is given by (see Fig. 17-286) 

C(s) = K v 
R(s) K v + s 


The error function is given by (see Fig. 17-28c) 

E(s) s (1 fK v )s 

R(s) K v +s 1 + (s/K v ) 

The response of a Type 1 system step input is given by (see Fig. 17-29) 


r(t) = Ah(t ) R(s) = — 


C(s) = 

E(s) = 


s[l + s/K d1 
IIK V A 


c(t ) = A(1 — e K v*) 


e(t) = Ae K v* 


(1 + s/K v ) 

A Type 1 system does not develop an error in response to a displacement input. 


























RESPONSE 


690 


CONTROL SYSTEMS 



Fig. 17-29. Response of Type 1 system to step input. 


The response of a Type 1 system to ramp input is given by (see Fig. 17-30) 


r(t) = ant R(s) = ~- 

s 2 


C(s ) 


CO 

S 2 [l + (s/K d )] 


(Oj 

c(t) = cot - 

Kv 


(1 - e -* v ‘) 



Fig. 17-30. Response of Type 1 system to ramp input. 




















RESPONSE 


LINEAR SYSTEMS 


691 


A Type 1 system develops a displacement error a >i/K v in response to a ramp input ant. 
The response of a Type 1 system to parabolic input is given by (see Fig. 17-31) 


r{t) = 


att 2 

~2~ 


R(s) 


a,_ 

s 3 


C(s) 

a. 

c(t ) 

s 3 [l -1- (s/Kv)] 

E(s) 

ai/K v 

e(t ) 

s 2 [l +{s/Kv)] 


ait 2 ai (^ 1 — 


a i 

K v 


2 K v V K v 

1 - e~ K v‘ 


( 1 — e~ K v l \ 

V - kt) 




Fig. 17-31. Response of Type 1 system to parabolic input. 


A Type 1 system develops a ramp error 



in response to a parabolic input. 

Notes on Behavior of Type 1 System 


(а) The response is identical to the simple RC network. 

(б) The time constant is 1/K V . 

( c ) The open-loop gain is unity at co c = K v ; the loop is often referred to as closed to co c . 

id) To the first order, letting t 1/w (increasing time implies decreasing ca), at 
t = 1 IK t -(aj = K v ), c(t) merges smoothly with r(t). 

(e ) The system gain at t — 1/co is of the order of the frequency-response gain, and the 
error input signal is reduced by this gain at this time. 

(f) The straight-line approximation to the system error shows that as time proceeds 
toward t ~ l/oo = », the error is r(t) until t = 1/K v (a) = K v ); i.e. the system has not 
responded until this time. 













692 


CONTROL SYSTEMS 


(g) Note that c(t) =K v Ae~ K v t = K v e(t ); i.e., in a Type 1 system the output rate is 
proportional to the error. 

Ui) The Type 1 system has an open-loop gain of K v at cj = 1, and a gain of unity 
at oj /C y. 

(i) A Type 1 system reduces the resultant error by one order of t\ i.e., no error for 
displacement input; a displacement error for velocity input, and a velocity error 
for parabolic input. 

17.1.4.4. Type 2 System (Many Tracking Loops, Gearless Torqued Loops). The open- 
loop transfer function of a Type 2 system is given by (see Fig. 17-32a) 

C(S) Kad+TS) 

- (T<<1) 





Fig. 17-32. Type 2 system Bode plot, (a) Open loop. 
(6) Closed loop, (c) Error response. 




















LINEAR SYSTEMS 


693 


The closed-loop transfer function of a Type 2 system is given by (see Fig. 17-326) 


and (see Fig. 17-32c) 


C(s) _ K a (l + ts) 
R(s) s 2 -f K a TS + K a 

E(s ) _ s 2 

R(s) S 2 + KaTS + K a 


The response of a Type 2 system to step input is given by (see Fig. 17-33) 


r(t) = A R(s) = — 

s 


C(s) = 


AK a ( 1 + ts) 

S(S 2 + K n TS + K a ) 


c(t) = A 


1 -Ne 


-KaT 
2 


t 


where 


1 


N = - 


sin (Ot -I- tp) 


[(KaT) 2 - 4 K a ] > 0 
[(KaT) 2 - 4 K a ] < 0 



K a T 2 \V' 2 


(frequency of oscillation) 


E(s) 


As 

S 2 + KaTS + K a 


e(t) = NA e~ K a T ‘l* 




Fig. 17-33. Response of Type 2 system to step input. 
























694 CONTROL SYSTEMS 

The response of a Type 2 system to ramp input is given by (see Fig. 17-34) 


(Oi 


r{t) — oat R(s) = — 
C(s) 


C Oi K a ( 1 + TS) 


S 2 (S 2 + K a TS + K a ) 


E(s) = 


(Oi 


s 2 4- KqTS + Ka 


c(t) = (Oi(N e K a Tt < 2 ) 


e(t) = Nojj e K a Ttl2 




Fig. 17-34. Response of Type 2 system to ramp input. 

A Type 2 system develops no steady-state error in response to a ramp input. 
The response of a Type 2 system to parabolic input is given by (see Fig. 17-35) 


r(t) = at 2 /2 R(s ) = 

do (s) 


a 


aK a ( 1 + ts ) 


s 3 (s 2 + K a rs 4- K a ) 


e(s) = 


a 


S(S 2 + K a TS + K n ) 


do(t) = - at 2 - — (1 - N e- K <* Tt ' 2 ) 
2 Ka 


a 


e(t) = — (1 - N <?-*«"/*) 

K a 











SAMPLED-DATA SYSTEMS 


695 




Fig. 17-35. Response of Type 2 system to parabolic input. 


A Type 2 system develops a displacement error a/K„ in response to an acceleration 
(parabolic) input. 

Notes on the Behavior of a Type 2 System 

If [{K a ry - 4K a ] < 0, the response is a sharp oscillatory rise with overshoot. The 
transient decays with a time constant of l/2c o c where o> c is the crossover frequency. 
The initial response of this system is faster than that of a Type 1 system of the same 
closed-loop frequency (see Sec. 17.1.4.3), but the transient decay time is longer. Again, 
the time domain performance can be crudely approximated from the plot (t ~ 1/oj). 

17.2. Sampled-Data Systems [3] 

17.2.1. Basic Definitions 

17.2.1.1. Sampling. The sampler modulates a series of periodic narrow pulses with 
the incoming, continuous signal. It is assumed, in general, that the pulses have zero 
width. These can be represented as 8{t — nT ), for which the Laplace transform is 

1 

(1 - e~ Ts ) 


A (s) = 















696 CONTROL SYSTEMS 

where T is the sampling period. If the input to the sampler is rit), the output is 


oo 

r*(t) = ^ r(nT)8(t — nT) 

n = o 

The transform relationships between the input and output can be written two ways: 

oo 

R*is) = 2 r(nT)e~ nTs 

n = 0 


R*(s) = ^ Yj R(s+jn(o s ) o) s = l/T 

n = oo 

The first equation is most useful in analysis; the second indicates the modulation char¬ 
acteristic, i.e., a series of carriers at ruo s , each modulated by rit). The frequencies 
around na) s are called complementary signals or sidebands. 

Two important sampling theorems are: 

(a) If the sampling rate is at least twice the highest frequency component, a signal 
can be transmitted without loss of information. 

ib) The mean square value of the samples pikT) is equal to the mean square value 
of the signal pit). However, this is true only if the signal contains neither any fre¬ 
quencies n<x) s l2 nor components such that 

| COi ± Cl)2 | = net)* 

17.2.1.2. Types of Sampled Systems. Two types of sampled systems are as follows: 

(а) Error sampled: The sampler follows immediately after the error-sensing device. 

(б) Nonerror sampled: This system can usually be reduced to an error-sampled system 
by applying rules of Sec. 17.2.2. 

17.2.1.3. Clamping or Holding 

Introduction. In general, clamping or holding circuits act as phase-lag circuits; 
the higher-order the hold, the more lag. They are used to remove complementary 
frequencies caused by the sampling process, convert the samples into approximately 
continuous form, supply energy for the actuation of the output device, and provide 
a finite drive for the continuous portion of the system. 

Types of Holding Circuits 

Zero-Order or Boxcar 

This circuit holds the measured value for part or all of the period between pulses: 

Cnit ) = c(nT) 

A full clamp holds the signal value until the next sample pulse. This produces less 
distortion but delays the signal by 77 2. The transfer function is given by 

G h0 (s) = (1 — e~ Ts )/s 

First-Order 

This circuit approximates the signal between pulses as a first-order polynominal: 

Cnit ) = cinT) + c'inT)it — nT) 




SAMPLED-DATA SYSTEMS 


697 


i.e., the slope is based on previously measured values, hence little error is generated 
for constant slope input. The necessity of obtaining an estimate of the slope causes 
an additional delay. The transfer function of this holding circuit is 


Ghi — 


1 + Ts fl - e~ 


Ts 


&th Order 


The general fcth-order holding circuit approximates the signal between pulses by 
a &th order polynomial: 

c n {t) = c(nT) + c’(nT){t — nT ) + • • • 
c k (nT)(t - nT) k 

The result of such approximation is more effective smoothing, but a greater delay is 
encountered. This system is usually detrimental to stability, as well as complex 
and expensive. 

17.2.1.4. Weighting Sequence g n (t). Output of a sampled-data system for unit im¬ 
pulse input (at t = OT) 

gn(t) = ginT) = g(t)\ t=nT 

00 

Note that g*U) = ^ g n (t)8{t — nT) 

n = o 

17.2.1.5. Equivalent Continuous Data Functions [3]. Table 17-1 gives some equiv¬ 
alent continuous data functions. 


Table 17-1. Equivalent Continuous Data Functions [3] 


Continuous-Data System 


1. Input and output: 


Pulsed-Data System 


Sit) 

yit) = wit) 

y n = Wn 

uit) 

yit ) = [ w(t) dr 

It 

yn = y W k 


Jo 

r‘ 

k = 0 

n 

xit) 

yit ) = 1 wiT)xit — r) dr 

y n = Y Wk Xn-k 


Jo 

rt 

k = 0 

n 


yit ) = 1 xir)wit — t) dr 

yn = y. Xk Wn-k 


Jo 

k = 0 

Frequency 

Yijio) = Aiw) 4 - jBito) 

Y*ij<o) = A*i<o)+jB*io>) 

functions 

f x 

X 


■A(o>) = yit) cos (ot dt 

= Y yinT) cos nTd) 


Jo 

n = 0 


B(oi) — — [ yit) sin u>t dt 

X 

B*icj) = — Y yinT) sin nTa> 


Jo 

n = 0 

Time 

functions 

1 r 

yit)— - I Aid)) cos toj da) 

2 n Jo 

yinT)=—[ A*id) cos nd dd 
* Jo 

yinT) = — -^J B*(0) sin n0 < 


f m 

X 

Stability 

condition 

I |u/(<)| dt < « 

Jo 

X W < 00 

n = 0 

Described by 

Differential equations 

Difference equations 





698 


CONTROL SYSTEMS 


17.2.1.6. z-Transforms. The system considered is shown in Fig. 17-36. 2 -trans- 
forms permit the relationship between R(s ) and C(s) to be written as 


R(z) 

C(z) 


= G{z) 


where z = e Ts and G(z) is the pulsed transfer function or 2 -transfer function for the 
system. 


R(s) 

R*(s) 


1 

r(t) 

/ r*(t) 

G(s) 

1 

i 


U 

g(t) 



Q) 




Q. 


S 


/ 

u, 

QJ 

a 

S 


cS 

CO 


CO 


C(s) 

c(t) 


C*(s) 
c* (t) 


Fig. 17-36. Basic sampled data system. 


Some basic relationships are: 


R(z) = R(j In 2 ) = 2 

' ' n = 0 


r(n, t)z~ n 


R(z) = z{r(t)} = L{r*{t)} , 
C(z) = z{G(s)i?*(s)} 

A list of 2 -transforms is given in Table 17-2. 
Some theorems relating to 2 -transforms follow: 

(а) They are linear 

(б) z{g(t — nT)} = z~ n G(z) 
z{g(t + nT)} = z +n G(z) 

if n > 0 and g(kT) = 0 for 0 < k < (n — 1). 


^ In z = z{ft(s)} 


(c) z{e at g{t)} = G(e~ aT z) 


z{e~ at g(t)} = G(e aT z) 


(d) lim f--- G(2)| = lim {g(0} 

z —* i L z J t -* x 

(e) lim {G( 2 )} — lim {g(0} 

Z —* oo 1—^0 

The following lists some general properties of 2 -transforms. 

(a) They are periodic, with period o> s . 

(b) G(z) is real at z = 1(oj = 0) and z = l(oo = nu) s /2). 

(c) The poles of G(z) are zr — e Ts k. 

( d ) Shifting the zeros of G(s) changes the gain constants in G(z). 

( e) If G(s) has simple poles only and no pole at the origin, G(z) describes a closed 
curve as z traverses the unit circle of the z plane. 

if) If G(s) has a pole at the origin, G{z) closes clockwise at infinity; i.e., there is 
a pole at z — 1. 










SAMPLED-DATA SYSTEMS 


699 


Table 17-2. z Transforms [3] 


G(s) 

1. 1 

2 . e- kT ‘ 

1 


s - — In a 


4. I 
s 


5. i 


git) 

m 

8{t — kT) 

ntIT 


uit) 


1 or z~° 


z — a 

z 

2—1 

Tz 

iz ~ l) 2 


G(z) 


g(nT) 

S(nT) 

8(nT - kT) 
a" 

u(nT) or 1 
nT 


6 - ? 


2 ! 


T 2 z(z + 1) 
2 ( 2 - l) 3 


2 (nTY 


7 - 7^ 


1 


(* - D! 


t k ~ l lim 


(-!)*-• d k ~' 


a-*0 (k 1)! da* 




a*-> 

(-1)*-' lim ( 

a -• 0 t)a 1 


8 . 


s -I- a 


.-af 


2 — e“ ar 


-anT 


10 . 


(s + a) 2 

1 

(s + a)* +1 


te 


-at 


k\ 


■ -at 


Tz€~ aT 
(2 — e~ aT ) 2 


-P.JLJ1 'I 

£! da* \2 — e _ar / 


(nT)€~ anT 




11 . 


s(s -(- a) 


1 - 


a/ 


(1 - €~ qr )2 
(2 — 1 ) (2 — € _aT ") 


1 - €- 


anT 


12 . 


s 2 (s + a) 


1 - 


Tz 


(1 - €- ar )2 


( 2 - 1) 2 0(2 - 1 ) (2 - €-“ 7 ') 


nT 


_ c-anT 


1 - € 


13. 


(Do 


s 2 + a»o 2 


sin (oat 


z sin (o 0 T 


Z 2 — 22 COS (OoT 4- 1 


sin n (OoT 


14. 


s 2 4- a>o 2 


COS (Oat 


z(z — cos (OoT) 
z 2 — 2z cos (o 0 T 4- 1 


cos n (o 0 T 


15. 


(O 0 


s(s 2 + Oi 0 2 ) 


1 — COS (Oat 


2(1 ~ COS (OpT) (2-1-1) 

(z — 1) (z 2 — 22 COS (OaT + 1) 


1 — cos n (OoT 


16. 


a> 0 


(s 4- a) 2 4- coo 2 


e~ al sin co 0 t 


ze~ aT sin a> 0 T 


2 2 — 2e- aT z cos (OoT 4- € _2a7 ^ 


e~ anT sin n a> 0 T 


g-aTn) 



































700 


CONTROL SYSTEMS 


The 2 -transform maps the left half plane into the inside of a unit circle, and the 
negative real axis into the positive real axis. 

17.2.1.7. Inverse z-Transforms. The basic method for finding the inverse transform 
is to apply 


ginT) = —— (f G(z)z n ~ 1 dz 
2nj J r 

% 

where T is the path enclosing all singularities of G(z)z n ~ 1 . 

The inverse transform can often be evaluated more easily by using 

ginT) = ^ (residues of G(z)z n ~ 1 at Zk) 

*k 


Another method is to perform a partial fraction expansion; then a table of trans¬ 
forms can be used to yield an approximate output. 

Still another method is the use of a power series expansion (long division): 


Division Yields 


G(z) = 


b m z m + b m ~iz m ~ 1 • • • bo 
C„Z n -f Cn- i2 n_1 • • • Co 


n 2= m 


Giz) = a 0 + ai 2 -1 -I- a 2 z~ 2 t • • • 


where the a* are the values of g*it) at the ith sampling period. 


17.2.2. Determination of Transfer Functions 

17.2.2.1. Transfer Function in s-Plane. For the open-loop system of Fig. 17-36, the 
transfer function is determined as the ratio of input and output Laplace transforms, i.e., 


Gis) 


Cis) 

R*{s) 


G*(s) 


C*(s) 

R*(s) 


For the closed-loop system (Fig. 17-37) the following can be easily derived: 


E(s) = R(s) — Bis) 


E*(s) = R*(s)-B*(s) 

B(s) = Gis)His) = GHis) 

E*is) = R*is) — GH*is) 

Cis) = E*is) Gis) 

E*is) 1 

R*is) ~ 1 + GH*is) 

Cis) Gis) 

R*is) 1 + GH*is) 

GH*is)=— GH(s+jnu) s ) 

n = -x 

17.2.2.2. Transfer Function in z-Plane. Derivation of the transfer function from 
Gis) can be accomplished by taking the inverse transform to obtain git). Then, sub¬ 
stituting t—nT will permit the application of 


Giz) = ^ ginT)z~ n 

n = 0 










SAMPLED-DATA SYSTEMS 


701 



Fig. 17-37. Sampled data feedback system. 


or, for complicated G(s), partial fractions can be used. Some 2 -transforms are given 
in Table 17-2. 

In the block diagram in Fig. 17-38, quantities or blocks not separated by a sampler 
are combined, and the z- transform of the combination taken. 

C(z) = G 2 {z)G\R(z) 



C(z) = RG^zJG^z) 


R(z) / 



C(z) = R(z)G 1 (z)G 2 (z) 


Fig. 17-38. Block diagram transfer-function derivation. 


Quantities or blocks separated by a sample are transformed and then combined: 

C(z) = G l (z)G 2 (z)R(z) 

The rules above can be applied for feedback systems, or the system output can be written 
in terms of the Laplace transform. Then let G(s) —*• G(z); G(s)H(s ) —► GH{z). 

17.2.3. Methods of Analyzing Sampled-Data Systems 
17.2.3.1. Directly from Differential Equation 
Representation of the System. In Fig. 17-37, let 

A(z) = GH(z) 

The characteristic equation F(z) is then the numerator of 1 + A (z). 

Absolute Stability Analysis. If F(z ) is a quadratic polynomial with real coeffi¬ 
cients, and the coefficient of z 2 is 1, the system is stable when all three of the following 
conditions hold: 


|F(0)| < 1 
F(l) > 0 
F(—1) > 0 

























702 


CONTROL SYSTEMS 


The Schur-Cohn Stability Criterion 
Write the characteristic equation as 

F(z ) = a 0 + a x z + o 2 z 2 ■ • • a n z n = 0 


Then form the determinant 


do 

0 

0 

• 0 

d n 

On —1 

• • • dn-fr+1 

a 1 

do 

0 

• 0 

0 

On 

* ’ ’ CLn-k + 2 

• 

• • 

do • • 

• 



• 



. . . 




• 

a-k-i 

dfc-2 

ax -3 * • 

• do 

0 

0 

* * * On 

On 

0 

0 

• 0 

do 

dl 

• * * dfc-1 

dn-i 

dn 

0 

• 0 

0 

do 

’ • * O-k — 2 

• 

• • 

dn ' ' 

• 



• 

dn-Ac+1 

dn —Ac + 2 

'i' 

On 

0 

0 

* * * do 


where k = 1, 2, • • • n 


Example for k = 3: 


a„ is the conjugate of a„ 


do 0 0 a 3 o 2 ai 
di a 0 0 0 a 3 d 2 
d 2 ai a 0 0 0 a 3 
a 3 0 0 a 0 di d 2 
d 2 a 3 0 0 ao ai 

dl d 2 d 3 0 0 do 


A system is stable if A* < 0 for k odd 

> 0 for k even 

17.2.3.2. Nyquist Method 

Representation of the System. Construct a polar plot of G(z ) as 2 goes once around 
the unit circle. 

Absolute Stability Analysis. The number of encirclements fl of the critical point 
(—1 or — 1/k ) indicates stability, since if fl = n — p where p = number of poles of G(z ) 
and n = number of zeros of G(z), the system is stable. 

Transient Response Analysis. The transient behavior of a sampled-data system 
can be determined in a manner similar to that used in continuous systems. 

17.2.3.3. Bilinedr Trdnsformation Anolysis. Make the substitution in G(z) 

1 + W 


This yields an expression which is the ratio of two polynomials of the same order in 
W. Continuous data system analysis techniques (such as Routh, Nyquist, Bode, and 







NONLINEAR SYSTEMS 


703 


root locus) can then be applied to determine stability, relative stability, etc. The 
open-loop transfer function in IT of a sampled-data system is nonminimum phase; 
both gain and phase must be sketched in applying the Bode method. 

17.2.3.4. Root Locus Method. The construction of root loci is carried out as for 
continuous systems, but in the z-plane. Table 17-3 indicates the expected behavior 
of a system from the pole location. 


Table 17-3. Effect of Pole and Zero Location 
on Behavior of Sampled-Data System [31 


Location of Closed-Loop Poles 

Outside the unit circle 
Inside the unit circle 

(а) Real pole in right half 

(б) Real pole in left half 

(c) Conjugate complex poles 


Mode of Transient Behavior 

Unstable operation 
Stable operation 

(a) Decaying sequence 

( b) Alternating decaying sequence 

(c) Damped oscillatory sequence 


17.2.4. Types of Sampled-Data Systems. In general, the characteristics of con¬ 
tinuous-data systems carry over to these sampled-data systems (see Sec. 17.1.4). 

Type 0: No Poles at z = 1 

(a) Position error = -—— 

1 + G(z) 


(6) Velocity error = °° 

(c) Acceleration error = °° 
Type 1: One Pole at z = 1 


(a) 

Position error = 

0 



Velocity error = 

T{z — 1) 

T 

(6) 

G(z) 

~ K, 

(c) 

Acceleration error = 00 



Type 2: Two Poles at z = 1 

(a) Position error = 0 

( b ) Velocity error = 0 


(c) 


Acceleration error = 


THz- l) 2 
G(z) 



17.3. Nonlinear Systems 
17.3.1. Basic Definitions 

General Characteristics of Nonlinear Systems 

(а) Superposition does not hold. 

(б) Sinusoidal inputs do not necessarily produce sinusoidal outputs. 





CONTROL SYSTEMS 


704 

(c) System stability may depend on input amplitude or frequency. 

( d ) Instability may be exhibited by the presence of limit cycles, i.e., oscillations of 

constant amplitude and arbitrary waveform. 

(e) Subharmonics can be generated. 

Phase Plane. A graphical determination of system performance in terms of state 
variables (output or error position, velocity, etc.). 

Singular Point. A singular point is a point of unusual behavior in the phase plane. 
Structural Stability. If the system is described by 

x = P(x, y ) 


y = Q(x,y) 

small perturbations of the coefficients P and Q will not cause a change in the behavior 
of a structurally stable system. (Saddle points and centers are characteristic of struc¬ 
turally unstable systems.) 

Stability (Around an Equilibrium Point) 

(а) Stable in Liaponov sense: The state variables are bounded in a region for any initial 
disturbance in that region, and the upper bound is a function of the initial disturbance. 

(б) Asymptotic stability: In addition to being stable in the Liaponov sense, the system 
approaches the equilibrium point asymptotically. 

(c) Global stability: The system is asymptotically stable over the entire phase space. 

Limit Cycle. A limit cycle is an oscillation of constant amplitude and arbitrary 
waveform. There are two basic types of limit cycles. 

(1) Stable limit cyple: If a solution is started inside or outside the limit cycle, the 
system will approach the limit cycle. A stable limit cycle must enclose either an un¬ 
stable limit cycle or an unstable equilibrium point. 

(2) Unstable limit cycle: If a solution is started inside or outside the limit cycle, 
the system will diverge from the limit cycle. An unstable limit cycle must enclose 
a stable limit cycle or a stable equilibrium point. 

Hard Excitation. Hard excitation is excitation such that the system is outside 
an unstable limit cycle. 

Characteristic Exponent of Limit Cycle. If n 0 is the perpendicular distance of 
a phase-plane path from the limit cycle, and n is the perpendicular distance of the path 
one cycle later, in the neighborhood of the limit cycle, 

n = noe a <> 

a„ is the characteristic exponent; if d 0 < 0, then there is a stable limit cycle; and if 
a () > 0, then there is an unstable limit cycle. 

First Canonic Form of Differential Equations. Consider the system of Fig. 17-39. 
Let the following hold: 

r = 0 (no input for t > 0) 

y = fix) (description of nonlinear element) 


Then the system nonlinear element can be described by 


dzj 

dt 


h-iZi 


+ fix) 


i = 1,2,. . .,n 



NONLINEAR SYSTEMS 


705 



Fig. 17-39. Basic nonlinear feedback system. 


n 

X = 2 OCiZi 
i = 1 


— = 2 0*. - '■/’(*) 

where are the poles of Gi(s)G 2 (s) 

a, are the negatives of the residues of Gi(s)G 2 (s) 

n 

Pi aikir = — a; i = 1 , 2 ,.. n 

i= 1 

Note that this description is valid only for systems in which the poles of Gi(s)G 2 (s) 
are simple and in which the number of zeros of Gi(s)G 2 (s) is less than the number 
of poles. 

17.3.2. Methods of Analyzing Nonlinear Systems 

17.3.2.1. Time-Domain Analysis. For systems whose nonlinearity can be simply 
described by low-order polynomials, ( e.g ., straight-line approximations or simple 
curves), the system performance can be determined in a piecewise fashion. First 
write the system equations for operation in each mode of nonlinearity behavior. Then, 
for the given input, obtain a piecewise solution, using the system equation appropriate 
to the nonlinear behavior. To evaluate constants, use the output value at the end of 
each time interval as the initial condition for the interval which follows. 

17.3.2.2. Consideration of the Differential Equation 

Bendixson’s Theorem. The system equation is written in terms of the state vari¬ 
ables, i.e., 


x = Pix, y) 


Then if 


y = Q(x,y) 


dP j dQ 

dj- d y 


* 0 


and is of constant sign in a region, there is no limit cycle in that region. 

Existence of Limit Cycles for Second-Order System. Consider the equation 


x + f(x)x + g(x) = 0 

The following are sufficient but not necessary conditions for the existence of limit cycles. 

(а) fix) and gix) are analytic (can be expressed as power series) 

(б) -gix) = gi-x) 

(c) xgix) > 0 














706 


CONTROL SYSTEMS 


(c0 f(x) = f(-x ) 

(e) AO) < 0 

Define the functions 


F(x) = 



dx 


then: 

if) Fix) as x —> 


Gix) = 



dx 


ig) Gix) — 0 has a unique root at x = a, where a > 0, and is monotone-increasing 
for x > a. 


17.3.2.3. Describing Function Analysis 

Introduction. In applying this method it is assumed that there is only one non¬ 
linear element in the system, that the nonlinear element is time invariant, and, if 
the input is a sinusoid, that the only significant component of the output is the funda¬ 
mental component; no subharmonics are generated. 

Describing functions are defined by 


N = 


Amplitude of the fundamental at the output, c 
Amplitude of input sinusoid, r 


Ci 

ri 


The third harmonic of the output usually causes the largest error [2], hence |c 3 /ci| 
measures the accuracy of the describing function in predicting the true output. 

Open-Loop Describing Function. The open-loop describing function N 0 can be 
found by first determining the nature of the nonlinearity and then hypothetically 
(or experimentally) inserting an input c = X sin a >t. Then N 0 is the plot of amplitude 
of fundamental of the output vs. X. (For examples of open-loop describing functions, 
see Fig. 17-40 through 17-43.) 




Fig. 17-40. Open-loop describing function for nonlinear element 
with dead zone, (a) Description of nonlinear element behavior. 
(6) Describing function. 










NONLINEAR SYSTEMS 


707 




(a) (b) 

Fig. 17-41. Open-loop describing function for nonlinear element with saturation, 
(a) Description of nonlinear element behavior. (6) Describing function. 




(a) (b) 

Fig. 17-42. Open-loop describing function for nonlinear element with dead zone and saturation, 
(a) Description of nonlinear element behavior. (6) Describing function. 



I 

max 



(a) ( b > 

Fig. 17-43. Open-loop describing function for nonlinear element with hysteresis, 
(a) Description of nonlinear element behavior. (6) Describing function. 
































708 


CONTROL SYSTEMS 


Stability Analysis from Describing Functions. If the system transfer function 
is KG(s), make a Nyquist plot of G(s). Consider the gain to be N 0 K. Then plot the 
critical point locus, —l/N 0 K, where iV 0 may include phase shift. (See, for example 
Fig. 17-44.) If No is frequency-dependent, several critical point loci must be plotted. 




Fig. 17-44. Example of stability analysis using 
describing functions, (a) Stable for all input 
signals. (6) System with limit cycles: for input 
amplitudes less than a, the system is stable. For 
input amplitudes greater than a, the system has 
a stable limit cycle at b. 


Closed-Loop Describing Function N c . N c can be found as follows (see Fig. 17-45): 

(a) Assume a frequency o> and calculate KG( w) 

( b) Assume an error voltage and determine y from the describing function, N 0 . 

(c) The closed-loop system gain at co is then 

N 0 KG(j(o) 

G c (jco) = -—:— /. —r = Me jN 


1 + N 0 KG(j(o) 


C d ) Co = EN 0 KG{j(i)) 

Ri = (1 + N 0 KG(j(t))E e j(ut 
N 0 KG(ja>) 


N — 


No 


E 















NONLINEAR SYSTEMS 


709 


(e) Repeat for different E’s and plot M and N vs. input amplitude A. 

(f) Repeat for different w. 



Fig. 17-45. Closed-loop system for describing 
function analysis. 


Example of Closed-Loop Describing Function. Figures 17-46 and 17-47 show 
the closed-loop describing function for a second-order system with torque saturation. 



Fig. 17-46. Magnitude M of closed-loop describ¬ 
ing function of second-order control system for 
nonlinear element with saturation. 



Fig. 17-47. Phase shift N of closed-loop describ¬ 
ing function of second-order control system for 
nonlinear element with saturation. 


17 . 3 . 2 . 4 . Phase-Plane Analysis 

Introduction.. Phase-plane analysis is most useful for second-order systems; it can 
be used to study transient behavior subject only to initial conditions (i.e., no other exci¬ 
tation); only time-invariant systems can be considered. The performance is plotted 
with state variables as the coordinates. In general, this technique is not applicable 
to systems of higher order than 2 or 3, but by extending the concept it can provide 
insight into the performance of systems of any order. 

Obtaining Phase Portrait 

Direct Solution of Differential Equations (for Linear Systems). The differential 
equation can be solved to obtain x, which is then differentiated to obtain x. t is then 
eliminated between x and x. Better results are usually obtained if the origin is moved 
to the maximum-velocity portion of the nonlinear characteristic. 

Solution of Differential Equation for x. y = x can be substituted into the equation and 
the equation divided by x. Noting that y/y —> dyldx, the new equation can often be 
solved. For example: 

x + (o„ 2 x = 0 
y -I- oj„ 2 x = 0 





















710 


CONTROL SYSTEMS 


y | 

y y 


= o 


dy , 2 * - n 

— -h (tin _ — 0 

dx y 


+ x 2 = K 2 

(tin 

The isoclines (lines through which the slope of the solution is a constant) are then 
determined. For example, consider x 4- a x x x + a 0 x = 0 

Let x = y 


y = — a x y — a 0 x 

y _dy dt _ dy _ a x y — a 0 x 

x dt dx dx y 


y = 


Qq 

s + b 1 


where s = dy/dx. 

This is the equation for the isocline on which the solution has slope s. Next, the 
direction of the solution can be determined, since x = y. If 3 / is positive, x is positive, 
and the solution moves toward the right in the upper-half plane. 

Types of Phase-Plane Plots. If, for a second-order system, Ai and \ 2 are the eigen¬ 
values, then the phase-plane plots shown in Fig. 17-48 are summarized as follows: 

Unstable node Ai >0, \ 2 > 0, Ai > \ 2 

Stable node Ai < 0, A 2 < 0 


Saddle point 
Unstable focus 
Stable focus 
Center 


AiA 2 < 0 

\ = p + iq p > 0 

k = p + iq p < 0 

Ai = A 2 = p + iq p = 0 


Time Response from Phase-Plane. The time response can be formed simply by 
forming 


t\ — t 2 — 



where y — dx/dt (the ordinate). Note that this equation can also be used in the error 
plane, where 

r e 2 •. 

t\ — t 2 = — — de 

Je t e 

17.3.2.5. Stability Analysis by Liaponov’s Second Method 

Introduction. This method can be applied to higher-order systems if the differ¬ 
ential equation is written in the first canonic form. However, this closed-loop system 
must have only one nonlinear element, and the nonlinear element must satisfy 

f f(a ) da 3s 0 * 

Jo 

for the closed-loop system with a single nonlinear element. 







NONLINEAR SYSTEMS 


711 








Fig. 17-48. Phase-plane plots for second-order system, (a) Stable node. (6) Unstable node. 

(c) Saddle point. ( d ) Center, (e) Unstable focus. ( f ) Stable focus. 

Liaponov’s Theorem. Consider the autonomous system 

yi = Y i (y 1 ,y 2 ,. . . ,y n ) 

If there exists a real-valued function V(y u y 2 ,. . .,y n ) such that. 

(а) V{y x , y 2 ,...,y n ) has continuous first partial derivatives 

(б) V is positive definite, i.e., V > 0 for any |y<| > 0, U(0) = 0 
(c) lim V(y 1 ,y 2> ...,y„) = « for all |y f | -*• « 

I Vi | - » 

then the equilibrium state of the system is as follows: 

(а) Stable, if there is some region V < k, (where k is some positive constant) such that 

o 

dt " dy t dt 

i.e., dV/dt is negative-semidefinite. 

(б) Asymptotically stable, if dVIdt < 0 in the region V < k, or if dVIdt 0 and 
dV/dt = 0 is not a nontrivial solution of 

yi = Yi (yi,y 2 ,.. .,y«) 

(c) Globally stable if condition ( b ) holds for the entire phase space. 

Liaponov Function for Systems Described by First Canonic Form: 

n n 

V = QjkYiYkCliic Q-ki 

i=l 1 




















712 


CONTROL SYSTEMS 


The stability equation becomes 

n 

2 a, ^ cij/(\i + \j) = at i=l,2,...,n 

j=i 

The system is asymptotically stable if 

(а) There is at least one solution such that the a, are real for real \,’s and are in com¬ 
plex conjugate pairs for corresponding complex conjugate pairs of A*. 

( б ) Re\i < 0 for all i = 1,2 ,..n 
(c) The nonlinear element satisfies 

xfix ) s* 0 , AO) = 0 

for all |;r| > 0 

More examples are found in [7], 

17.3.3. Specific Solutions 

17.3.3.1. Second-Order Conservative Systems 

General Comments. The equation describing a second-order conservative system is 

x + f(x) = 0 

Second-order conservative systems exhibit no damping and are described in the phase- 
plane by 

x = y 


or 


Potential Function. 


y = - fix) 

dy fix) 

dx y 

A function vix) can be defined as 


vix) = 



dx 


Hence the system equation becomes 


y 2 l 2 + vix) = h 

or 

kinetic + potential = total 
energy energy energy 

Behavior Near an Equilibrium Point. If the second derivative of vix) is taken, 
then, if 

d 2 vjx o) = dfjxp) 
dx 2 dx 


The system exhibits center behavior, if 


d 2 vjx o) = dfjxp) 
dx 2 dx 


The system exhibits saddle behavior; and if (see Fig. 17-49) 


d 2 vjx o) 
dx 2 


dfixo) 


= 0 


dx 









NONLINEAR SYSTEMS 


713 


v(x) = x^ + 5x^ + x 



Fig. 17-49. Construction of phase-plane plot 
of x + 3x 2 + 10* +1 = 0. 


Sketch of Phase Portrait [2] for Conservative Second-Order System. In sketch¬ 
ing the phase portrait, the singular points are located (maximums and minimums of 
v ) and the nature of each point determined from the preceding discussion. Then the 
lowest energy path is constructed, then the next higher, etc. (see Fig. 17-49). 

Effect of Varying a Parameter a 

(а) Consider the equation 

x + fix,a) = 0 

(б) Plot f(x, a) = 0, which is a plot of equilibrium points. 

(c) Shade the region where fix, a) > 0; then, for a given a, the equilibrium points 
are given by the curve. The number and type of equilibrium points can be determined 
as follows: if, as x increases, fix, a) changes from shaded to unshaded; the system 
exhibits saddle behavior; if the change is from unshaded to shaded, the system ex¬ 
hibits center behavior. 

17.3.3.2. Examples of Systems with Saturation 

Introduction. An error plane can be constructed as follows: 

(a) Write the system characteristic equation in terms of 


€ = 0i — 6 0 








































714 CONTROL SYSTEMS 

The linearity of this system depends on the linearity of the torque output T since 


e(t) = T[e(t), e(t), f e(t) dt,...] 

(6) Determine switching lines which divide the plane into regions of linear and 
nonlinear operation (it is assumed that nonlinearities can be approximated by straight 
lines). 

(c) Determine the qualitative behavior in each region. 

( d) Sketch the phase portrait. Note that the behavior of a saturated system is 
parabolic. 

Error Plus Error-Rate System with No Damping. Consider the system 

Ie(t) — n [e(t) -I- C e e(t )] 

where C e is the error-rate coefficient and the torque saturation curve is as shown in 
Fig. 17-50 (a). 



(a) 



Fig. 17-50. Phase-plane for system with torque 
saturation, (a) Description of nonlinear element. 
(6) Phase portrait. 








NONLINEAR SYSTEMS 


715 


If there is no acceleration input, I’rit) = 0 and 


from which 
and, integrating 


I'e(t) = — n[C e e(t) + e{t )] 
T = n[C e e(t) -f eU)] 


e(t ) = — — e{t) + 

^ e 


T 

fxC e 


Therefore, the switching lines are straight, representing torques of ±T m with slope 
— 1 IC e and abscissa intercept ±T m /ixC e (see Fig. 17-506). 

The behavior (see Fig. 17-506) is linear between the saturated torque lines and para¬ 
bolic on either side of this region. The phase portrait is sketched in Fig. 17-506. 

Bang-Bang System with No Damping. In this system the torque is described (see 
Fig. 17-51a) by 

T = B e(t)!\e{t)\ = ±B 


B 

Torque 



-B 

Input 




(a) 



Fig. 17-51. Phase-plane for bang-bang system 
without damping, (a) Description of nonlinear 
element behavior. (6) Phase portrait. 










716 

and the system equation is 


CONTROL SYSTEMS 


Ic(t) = B e(t)/\e{t)\ 

There is no region of linear operation; the switching line is the ordinate, and the 
behavior on both sides of the switching lines is parabolic (see Fig. 17-516). 

Bang-Bang System with Coulomb Damping. The torque is described by 

T = B e(t)/\e(t)\ - C e(t)l\e(t)\ 

(see Fig. 17-52a) so the system behavior is 

Ic(t) = B e(t)/\e(t)\ + Ce(t)/\e(t)\ 

Again there is no linear operation, and the behavior is parabolic in each quadrant: 

(а) Quadrant 1: T = B + C 

(б) Quadrant 2: T = —B + C 

(c) Quadrant 3: T=—B — C 

( d ) Quadrant 4: T = B — C 
(See Fig. 17-526). 


B 

Torque 



-B 

Input 




(a) 



(b) 

Fig. 17-52. Phase-plane for bang-bang system with 
coulomb damping, (a) Description of nonlinear 
element behavior. (6) Phase portrait. 








NONLINEAR SYSTEMS 


717 


Bang-Bang System with Viscous Damping. The torque is described as 

T = B e(t)l\e(t)\ + fe(t) 

(see Fig. 17-53); the system equation is 

rc(t) = B e(t)/\e(t)\ +fe{t) 

There is no region of linear operation, and the behavior is parabolic: 

(a) Quadrant 1: T = B + fe(t) 

(b ) Quadrant 2: T = —B 4- fe(t) 

(c) Quadrant 3: T = —B -I- fe(t) 
id) Quadrant 4: T = B + fe(t) 

That is, the system is highly damped for large error rates, and lightly damped for small 
error rates (see Fig. 17-53). 


Torque 


B 


Input 


-B 


(a) 



(b) 

Fig. 17-53. Phase-plane for bang-bang system 
with viscous damping, (a) Description of non¬ 
linear element behavior. (6) Phase portrait. 








CONTROL SYSTEMS 


718 


17.4. Design Methods 

17.4.1. Gain Adjustment 

Gain Adjustment from the Polar Plot. System performance can be strongly 
influenced by gain adjustment; the proper adjustment for desired operation* can be 
derived from using a Nyquist plot as follows: Draw a Nyquist plot of G(joj) [or of G*(ju )) 
for a sampled-data system]. Draw a ip line through the origin at ip = sin -1 (1 /M p ), 
where M p is the desired peaking of the transfer function. Draw a circle centered on the 
negative real axis, tangent to both the ip line and the G(j(o) locus. Determine K from 

K Ml 1 

c(AV - 1) d 

where c = the distance from the origin to the center of the circle 

d = the distance from the origin to the negative real-axis intercept of a perpen¬ 
dicular which passes through the point of circle and (//-line tangency. 

Gain Adjustment from Bode Plot. Proper gain adjustment is that which yields 
the desired gain and/or phase margin (usually about 30°). 

Gain Adjustment from log Modulus Plot. Variations in gain are reflected in 
a vertical shifting of the plot. Hence, using a Nichols chart, the gain which will result 
in tangency to the desired M circle is chosen by adjusting the vertical position of the 
plot. 

Comments on Gain Adjustment. An increase in gain decreases stability, de¬ 
creases steady-state errors, increases transient errors, and lowers response time. In 
general, the result of an increase in gain is higher M p , a>„, and o> p . 

17.4.2. Cascade or Series Compensation 

Simple Compensation Networks 

Phase Lead Network. This network (Fig. 17-54) provides positive phase shift. The 
transfer function is given by 


where a = RJ(Ri -f R 2 ) 


K c G c (s ) = a 


ST 2 + 1 
S(XT 2 + 1 


T 2 


= Ri 


C 2 


The Nyquist and Bode plots of this network are also shown in Fig. 17-54. The phase 
shift obtained as a function of a is also indicated. 

Phase Lag Network. This network (see Fig. 17-55) provides negative phase shift. 
The transfer function is 


K c G c (s) 


t 2 s + 1 
T 2 as + 1 


Figure 17-55 illustrates Nyquist and Bode plots as well as the obtainable phase shift. 

Design Using Nyquist Plot. A compensating network can be designed by the 
following. 

(а) Draw Nyquist plot and M circle for desired M p . (M p is approximately the 
transient response overshoot.) 

(б) Select resonant frequency a > p of compensated system. This is related to the 
speed of response (see Sec. 17.1.3). 

(c) If (d p is the frequency of peak of the uncompensated systems, then the network 
must yield a phase shift of 

<p = / Ocop' — / Q(Op 









DESIGN METHODS 


719 




Fig. 17-54. Phase-lead network. 


and a gain of 

(JL = Ooip' / 0(t) p 

where Oca/ is the distance from the origin to a> p ' and Oco p is the distance from the origin 
to a)p. Therefore, for a phase-lead network, Fig. 17-54 is used to determine a and for 
a phase lag network Fig. 17-55 is used to determine a; r 2 is determined from (6) above. 

( d ) The locus of the compensating network is plotted, using (r 2 , a) (or a) and /t x, 
as derived. 

(e) By vector multiplication, the new system locus is constructed. 

(/*) Repeat the steps above as required for different o> p and a (or a) until a satis¬ 
factory system is obtained. 

In sampled-data systems it may be desirable to use a pulsed-data network for com¬ 
pensation. In this case, the procedure is as outlined above, but the 2 -plane is used 
for design. 




























720 


CONTROL SYSTEMS 



<t> = tan 1 "/= - tan */a 

max v a 


a 

^max 

2 

-9.73 

3 

-24.68 

4 

-33.43 

5 

-41.87 

6 

-45.58 

10 

-54.9 



a 


Fig. 17-55. Phase-lag network. 


Design Using log Magnitude and Phase Plots. The method outlined below may 
be applied to both the log modulus and the Bode plots. 

(а) Draw the Bode or log modulus plot of the uncompensated system and decide 
what type of compensation network is required. 

(б) Determine the range of frequencies over which compensation is necessary. 

(c) Draw the plot for the compensating network. 

( d) Add the compensation plot to the system plot. 

(e) Adjust the system gain as necessary to achieve desired performance. 

( f) Repeat as necessary to obtain desired result. 

Other Compensation Schemes. Some additional compensation schemes follow. 





















DESIGN METHODS 


721 


First-Derivative Error Control. Although there is no change in steady-state error K p , 
the transient response is faster. Application of first-derivative error control does not 
change the order of system and increases K v without affecting the stability or phase mar¬ 
gin excessively. It does, however, affect the apparent viscous damping (see Fig. 17-56). 



Fig. 17-56. First-derivative error control system. 


First-Derivative Input Control. This method of compensation improves transient 
response, raises K p , and K v , and reduces the amplitude of the oscillations. It also 
affects the apparent viscous damping (see Fig. 17-57). 



Fig. 17-57. First-derivative input control system. Gi(s) 
is the controller. G 2 (s) is the controlled system. 


First-Derivative Output Control. This system can reduce steady-state velocity error 
(even make it negative). Since the nature of transient response is changed, caution 
should be used in applying it. Again, the apparent viscous damping is affected (see 
Fig. 17-58). 



Fig. 17-58. First-derivative output control system. 


Integral Control. This type of compensation reduces the steady-state error, which 
eventually becomes zero. Speed of response is increased but relative stability is 
decreased (see Fig. 17-59). 





Fig. 17-59. Integral control system. 








































722 


CONTROL SYSTEMS 


17.4.3. Root-Locus Method 

17.4.3.1. Introduction. The effect of adding poles and zeros to a system is indicated 
in Fig. 17-60. Real-axis zeros tend to spread the loci faster and stabilize the system. 
Real-axis poles, on the other hand, tend to make the loci spread more slowly and curve 
toward instability. 

The root-locus design method (after Guillemin in [2]) involves three steps: the closed- 
loop transfer function is determined, then the open-loop transfer function, and finally 
a compensation network is synthesized. 




(a) (b) 




Fig. 17-60. Effect of additional poles and zeros, (a) Effect of zero on locus with 
complex poles. (6) Effect of zero on locus with real negative poles, (c) Effect of pole 
on locus with complex poles. ( d ) Effect of pole on locus with real negative poles. 


17.4.3.2. Determination of Closed-Loop Transfer Function. Since the fixed elements 
of a system cannot be changed to bring about control, their characteristics must be 
determined by experimental or analytical techniques; these characteristics are then 
used to design an effective control system. 

















DESIGN METHODS 


723 

There are several basic relationships between the system specifications and the poles 
and zeros. If the system transfer function is written 


C(S) _ (S + Zi)(s + Z 2 )(S + Z m ) 

R ( s ) (s + Pi)(s + Pi){s + p „ ) 

Then the basic relationships between the system specifications and the poles and 
zeros are 

(1) (p — z) t 3* (p — z) c 

where p is the number of poles 


z is the number of zeros 


( 2 ) 


m 


K Y[Z, 

__ 

n m 

Y\p 1 -Kl\z ] 


where K p is the position-error constant; 


(3) 


1 


K v 




1 


Zt 


where K v is the velocity-error constant; 


(4) 



1 + i 1 


K v 2 


“ Pj-2 
j =i ^ 



where K a is the acceleration-error constant. 
Example: 


C(s) _ a),, 2 

R(s ) s 2 2a) n s+ (o n 2 


K p = 1 


K v = 


21 

(On 


1 (1 ~ 4 ^ 2 ) 

K a (O,, 2 

(5) Delay time (for step input) T rf ~ 1/K V (exact if g{t) is symmetrical around its 
centroid, i.e., the system exhibits small overshoot). 

(6) Rise time: (exact if g(t) is symmetrical around its centroid, i.e., the system ex¬ 
hibits small overshoot). 

Tr 2 2 1 

2 ~ Ka Kv 2 

Method for Compensation Using Poles and Zeros. The object of this compensation 
method is to base control primarily on two control poles, i.e., complex conjugate poles 
close to the imaginary axis. These poles and any significant zeros determine the system 
characteristics. In placing the control poles, other poles are placed either so far out 
from the imaginary axis, or so close, that they are not significant. In more complex 
systems, inner loops may have to be designed with other poles and zeros (see Sec. 17.4.5). 

The procedure described above ensures that the open-loop poles lie on the negative 
real axis; hence, the compensation can be accomplished with passive RC networks. 












724 


CONTROL SYSTEMS 


Potential Analogy for K v Adjustment. The object of this adjustment is to increase 
K v so as to reduce errors in response to velocity inputs. This adjustment should be 
made without materially affecting the relative stability of the system. 


If K p = —~ ^ i.e., = l), the velocity constant K v is analogous to the inverse of the 


electric field at the origin. Hence, poles are considered to be unit line charges (per¬ 
pendicular to s-plane) and zeros are unit line charges of opposite polarity. Therefore, 
the field at any radius r from a pole or zero is ±l/r. 

Using this analogy, zeros can be placed on the negative real axis in such a position as 
to reduce the field at the origin to 0; hence, an infinite K v is obtained. This is lead con¬ 
trol. 

Note that a pole and zero could be placed close together so their net effect is negligible 
on the other poles and zeros, but close to the origin so that the effect on the field at the 
origin is great. If the pole is outside the zero, K v is increased, while if the pole is inside 
the zero (closer to the origin), K v is decreased. The increase in overshoot produced by 
this compensation is PJZi , and, because of the term e~ Pit , a long-duration "tail” will be 
found in the response. 

17.4.3.3. Determination of Open-Loop Transfer Function. Once C(s)/R(s ) has been 
obtained, the open-loop transfer function G(s) can be formed by first writing 


Cjs) _ pis) Gjs ) _ pis) 

Ris) n(s) 1 4- G(s) pis) + q(s) 

Note that the poles of the open-loop system are the zeros of n(s ) — p(s); therefore, if 
the real part of n(s) is plotted on the same graph with the real part of pis), the roots 
of the open-loop system are the abscissas of the intersections of the two curves. 

Note that for most systems the sum of the zeros of qis) is identical to the sum of the 
poles of Cis)/Ris) ii.e., the zeros of nis)). 

17.4.3.4. Synthesis of Compensation Networks. The desired system response is 
obtained by using as many poles and zeros of the controlled system as possible and 
canceling the others approximately with a compensation network. 


17.4.4. Optimum Transient Response Behavior for Torque-Saturated Systems. 

Two of the torque-saturated curves pass through the origin (see Sec. 17.3.3.2), one for 
maximum positive torque, the other for maximum negative torque. These parabolas 
form the optimum switching curves (optimum implying the curve which takes the least 
time to reach the origin). Therefore, the fastest system will start heading toward 
the switching curve with maximum torque; when the switching curve is reached, 
the maximum opposite torque will be applied; this will take the system along the 
switching curve, to the origin. For small errors, or operation near the switching curve, 
a practical system based on this technique does not work too well. Therefore, a linear 
mode might be selected in these regions. The switching curve can be established using 
nonlinear elements. 


17.4.5. Miscellaneous Comments on Design and Compensation 

Crude Analysis. Since in a closed-loop system 

Cjs) KiGiis) 

Ris) 1 + KGris)K 2 G 2 is) 

where KiG x is) is the forward loop transfer function, and K 2 G 2 is the feedback loop 
transfer function, 

Cis) 1 


Ris) K 2 G 2 is) 











DESIGN METHODS 


725 


for frequencies where K ] G ] (s)K 2 G 2 (s ) > > 1 and 

C(s) 

R(s) ~ KlGi{s) 

for frequencies where K X G X (s)K 2 G 2 (s) < < 1. 

Inner Loop Gain Distribution. In two-loop systems, the inner loop gain should 
be distributed primarily in the forward loop {K t ), thus reducing the gain required in 
the outer loop. 

Multiple-Loop Analysis and Synthesis: One-Third Rule. Generally, the response 
of some loops will be made faster than others, in order to maximize inner loop response, 
€.g., a loop for stabilization in a track system. The faster loop will usually exhibit a 
peaking at its closed-loop frequency. Experience shows that the outer loop should have 
a closed-loop frequency no higher than one-third that of the inner stabilization loop. 

Serial or Sequential Analysis. If the crossover frequencies of individual loops 
are far enough apart, the inner loops can be analyzed independently for stability and 
response, and the result used in the analysis of the outer loop. 

Parallel Analysis. All loops having a common element are considered parallel. 
In Fig. 17-61 the common element is A, and thus the parallel loops are AB (loop 1); 
AD (loop 2); and ACE (loop 3). Then the output at any frequency is roughly that de¬ 
fined by the loop with the greatest gain at that frequency (or time). When considering 
the stability of the system, only the loop which predominates in that frequency domain 
must be considered. 



Fig. 17-61. Block diagram for parallel analysis. 


17.4.6. Statistical Design 

17.4.6.1. Introduction. Statistical design deals with the design of systems in which 
it is assumed that the input signals are from ergodic, stationary, random processes. 

The error criterion most generally chosen is that the mean square error be reduced. 
This criterion places heavy emphasis on large errors and, as a consequence, often leads 
to a system design of low relative stability. It is, however, very convenient to handle 
mathematically and in many applications yields adequate results. 

In designing a system statistically, it is often useful to know the probability distribu¬ 
tion function of the random process. This can be found by determining what percentage 
of time a signal exceeds various amplitudes. Differentiating the result yields the 
probability distribution. This procedure can, of course, be carried out graphically. 

If the mean-squared error criterion is chosen, signals are adequately described by 
correlation functions. These functions are defined by 

1 f* 

<Mt) = lim — fiit)fj(t + r) dt 

t X J 




















726 


CONTROL SYSTEMS 


If i = j, the result is the autocorrelation function; if i # j, the result is the cross-cor¬ 
relation function. The correlation function can be found graphically by sampling the 
function every a seconds and shifting the time axis to generate fit + r). Multiplying 
and averaging the results yields the desired function. The major characteristics of 
these functions are as follows: 

(1) 0 h(t) = 0n( t) 

(2) </>u(0) is the average power 

(3) 0 n (00) |0„(r)| =£ 0 n (O) 

(4) If flit ) contains periodic components, 0n(r) contains components of the same 
period. 

(5) If fiit ) contains a dc component, so does 0u(r) 

(6) </>n(r) contains no phase information 

(7) If (4) and (5) do not hold, 0n(r)—»0 as r—»<* 

(8) 0u (t) is the sum of the autocorrelation function of the individual frequencies 
which make up fiit) 

(9) 0n( t) corresponds to an infinite number of time functions 

(10) If fiit) = fait) + fait), then 

01l(r) = 0aa(r) + 0 oa(t) + 0i>a(r) + 0 M>(t) 

(11) 0aft(r) = 0 6a (— r) 

(12) If fait) and fbit) are uncorrelated (e.g., signal and noise), then 

01l(r) = 0aa(r) -f (pbbir) 

(13) <f>abir) * [0.«(O)0»(O)]»/* 
where ( ) indicates the ensemble average. 

The power-density spectrum $>jjij(o) is another important descriptor of signal and/or 
noise characteristics. <t>jjija)) can be found by using a spectrum analyzer, and has the 
following properties: 

(1) <I>iiO‘w) =F{0n(r)}= f 011 iT)e~ j(OT dr 

J x 

(2) <*>ii(» has no phase information 

(3) <1> 11 (jr a>) does not correspond uniquely to a time function 

(4) <Puij(o) is real 

(5) ^>n0'o>) = <I>ii(- j(o) 

(6) d>n(» > 0 

(7) <I>ii ij(o) contains impulses at frequencies contained in a periodic input. 

Physical realizability of a system G(s) can be determined as follows: 

(1) |G0'a>)| is realizable if the Paley-Wiener Criterion is satisfied: 

r iiogiG 2 (»n , , 

I -“--- a CD < 00 

Jo 1 + w 2 

(2) git) is realizable if git) = 0 for t < 0 and git) —*■ 0 as t —»<» 

(3) If Gis) is a ratio of polynomials, it is realizable if all the poles are in the left- 
half plane (but excluding <» and the j o> axis). 



BIBLIOGRAPHY 


727 


17.4.6.2. Design Principles. If physical realizability of the system is not considered, 
the optimum transfer function G op t(j o>) for minimizing the mean-square error can be 
found from 


G op t(j(t>) — 


<&s S (jo)) 


Ga(j(o) 


^ss(joj) -h 4>nn0'<i>) 

where <D„0 ‘g») = power density spectrum of signal 
®nn(jo>) = power density spectrum of noise 
Gd(j(o) = desired system transfer function 

If physical realizability is a consideration, the optimum transfer function G op t(s ) 
is given by 


where 


G op t(s) 


-i+ 


Shr(s)J <J>ij + (s) 


Gd(s ) 


<*>•■< = = <!>,« + (s)<J>,r(s) 

and where the superscript + means all critical frequencies in the left-half plane, and 
the superscript — means all critical frequencies in the right-half plane. 

It should be noted that this design is based on the assumptions that the mean-square 
error criterion is adequate, that signal and noise are stationary time series, and that 
a linear, series compensation is desired. The problems associated with this method 
are that <J> ss (s) is difficult to obtain, the derived system often has very light damping 
and is sensitive to parameter changes. 

References 


1. H. James, N. Nichols, and R. Phillips: 'Theory of Servomechanisms,” McGraw-Hill Book Com¬ 
pany, Inc., New York, 1947. 

2. J. Truxal: "Automatic Feedback Control System Synthesis,” McGraw-Hill Book Company, 
Inc., New York, 1955. 

3. J. Tou: "Digital and Sampled-Data Control Systems,” McGraw-Hill Book Company, Inc., New 
York, 1959. 

4. M. Gardner and J. Barnes: "Transients in Linear Systems,” vol. 1, John Wiley and Sons, Inc., 
New York, 1954. 

5. J. Truxal: "Control Engineers’ Handbook,” McGraw-Hill Book Company, Inc., New York, 1958. 

6. G. Thaler and R. Brown: "Servomechanism Analysis,” McGraw-Hill Book Company, Inc., 
New York, 1953. 

7. Z. Rekasius and J. Gibson: IRE, Trans. Auto. Control 7, 3 (1962). 

Bibliography 

Ahrendt, W., and J. Taplin: "Automatic Feedback Control,” McGraw-Hill Book Company, Inc., 
New York, 1951. 

Andronow, A. A., and C. E. Chaikin: 'Theory of Oscillations,” Princeton University Press, Princeton, 
New Jersey, 1949. 

Bellman, R.: "Stability Theory of Differential Equations,” McGraw-Hill Book Company, Inc., New 
York, 1953. 

Brown, G., and D. Campbell: "Principles of Servomechanisms,” John Wiley and Sons, Inc., New 
York, 1948. 

Chestnut, H., and R. Mayer: "Servomechanisms and Regulating System Design,” vol. II, John 
Wiley and Sons, Inc., New York, 1955. 

Evans, W. R.: "Control-System Dynamics,” McGraw-Hill Book Co., Inc., New York, 1954. 

Guillemin, E. A.: "Communication Networks,” vol. II, John Wiley and Sons, Inc., New York, 1935. 
Guillemin, E. A.: "The Mathematics of Circuit Analysis,” John Wiley and Sons, Inc., New York, 1949. 
Guillemin, E. A.: Proc. IRE Natl. Electron. Conf. 9, 1953 (1954). 

Grief, H. D.: Trans. AIEE 72, 243 (1953). 







728 


CONTROL SYSTEMS 


Kazda, L. F.. "Proceedings of Workshop Session in Lyapunov’s Second Method” (The University 
of Michigan, Ann Arbor, Michigan, 1962). 

Lienard, A.: Revue Generale de l’electricite 23, 901 (1928). 

Mulligan, J. H.: Proc. IRE 37, 516 (1949). 

Ragazzini, J. R., and L. A. Zadek: Trans. AIEE 71, 225 (1952). 

"Reference Data for Radio Engineers,” 4th Edition, International Telephone and Telegraph Corp., 
New York, July 1957. 

Seamans, R. L., Jr., B. P. Plasingame, and R. C. Clentson: J. Aeron. Sci. 17, 22 (1950). 

Valley, G. E., and H. Wallman: "Vacuum Tube Amplifiers,” McGraw-Hill Book Company, Inc., 
New York, 1948. 

Zadeh, L. A.: Proc. IRE 38, 291 (1950). 


Chapter 18 

SYSTEM DESIGN 


Sol Shapiro 

General Precision, Inc. 


William L. Wolfe 

The University of Michigan 


CONTENTS 


18.1. Design Procedures; Generalities. 730 

18.1.1. Example of Procedure for Gross Analysis. 730 

18.1.2. Sensitivity Analysis. 731 

18.2. Equations for Sensitivity Calculations. 731 

18.3. Optical System. 732 

18.4. Scanning Dynamics. 734 

18.5. Scanning Techniques. 735 

18.5.1. Rotating Wedges. 735 

18.5.2. Other Scanning Methods. 736 

18.6. Example of Design for a Search System. 737 

18.6.1. Requirements Analysis.. . . . 737 

18.6.2. Design Analysis. 737 

18.6.3. Definition of Primary System Parameters. 740 

18.7. Tracking System Design. 743 

18.7.1. Conception. 744 

18.7.2. Sample System. 746 

18.8. Mapping Systems. 750 

18.8.1. Requirements. 750 

18.8.2. Basic Sensitivity Equation. 751 

18.8.3. Background-Limited D*-Q Trade-Off. 753 

18.8.4. Sample System. 754 


729 

























18. System Design 


18.1. Design Procedures; Generalities 

The design of infrared systems requires the employment of many engineering approx¬ 
imations, compromises, and judgments about when to approximate and how to com¬ 
promise. This chapter provides some hints and some equations for guidance. It 
should be understood that not all of them necessarily apply in all cases. 

The formation of the problem usually involves specifications or at least judgments 
about the following; this constitutes requirements analysis: 

1. Function 

2. Volume of coverage 

3. Volume coverage rate 

4. Desired resolution 

5. Range 

6. Target type 

7. Background 

8. Operating altitude 

9. Desired output form 

10. Size, weight, reliability, etc. 

The design analysis steps can often be thought of as: 

1. Gross implementation 

2. Sensitivity analysis 

3. Scanning or tracking analysis 

4. Output analysis 

Sensitivity analysis and scanning analysis are intimately related. 

18.1.1. Example of Procedure for Gross Analysis. 

1. Make an estimate of entrance-aperture flux density from the target radiation 
based on blackbody radiation and some chosen emissivity — or a better quick 
estimate if available. 

2. Estimate background fluctuations in about the same way, choosing "good” 
values for emissivity and temperature changes and an instantaneous field 
of about 1 mil, or one matching the target. 

3. Calculate dwell times necessary from instantaneous field, total field, and 
required frame rate. 

4. See if these are compatible with existing optics, detectors, rotation rates, 
reticle spacings, etc. 

5. Either refine the design of item 4, or "bend” some of the constraints, or change 
some assumed system parameters drastically. 


730 


EQUATIONS FOR SENSITIVITY CALCULATION 


731 


18.1.2. Sensitivity Analysis. Stated in its simplest terms the problem is to cal¬ 
culate the ratio of the target signal to the total noise; the total noise can usually be 
calculated as the sum of the background noise and the system noise. Thus: 

1 . Calculate the spectral, temporal, spatial characteristics of the target radiance. 

2 . Calculate the attenuation by the atmosphere of this signal. 

3. Repeat items 1 and 2 for the background. 

4. Assume an optical system (usually ffl, 50% efficiency) and choose an appropriate 
detector. Calculate system noise and aperture size necessary for detection. 

5. If step 4 is successful, stop; if it is not, choose other combinations that are suc¬ 
cessful — possibly changing the number of detectors to decrease the noise band¬ 
width, etc. 

After this, specific parameters and special problems must be considered. 

18.2. Equations for Sensitivity Calculation 

Many workers prefer to calculate the noise equivalent input on a step-by-step basis. 
For those who do not, the equations for a field-mapping application (target smaller 
than instantaneous field of view) and for tracking (target filling the field and target 
less than full field) are given below. 

Mapping sensitivity 

A H «- 4 ^ 2 [ NEP' df 

to \fAAt J 


Target-detection sensitivity 
Target > field 


H(T,e,r ) > (K + 1 )H(T u ci,r) - KH(T 0 , e 0 , r) 

H(T\,€i,r) — H(To,e 0 ,r) s* kNEl 

Target < field 

AH(T, e, r) ^ cr 2 r 2 {k + l)H{T u e x , r) - [(r 2 r 2 (k + 1) - A] H(T 0 , e 0 , r) 

NEP (noise equivalent power) for scanning systems of this type using a single 
detector with detectivity D* is 


NEP = 


fd a 1 / 2 

D*T l i 2 


NEI (noise equivalent input) is 

NEP 4 F 1 V* 

(ttD 2 I 4 ) e 0 y Tre 0 y DD*T' 12 

For a system with N 0 equal detectors, the solid angle scanned by each detector 
can be reduced by the number of detectors; thus 

4 F a 1 / 2 

NEI =-777^-—: 

TTToyDD 










732 


SYSTEM DESIGN 


In the equations above, 

A = target area 
B — Planck function 
D = diameter of entrance pupil 
D* = detectivity 
F — focal length 
5 = focal ratio 
H = irradiance 
k = confidence constant 

K = ratio of target signal to background signal 
r = range 
T = temperature 

y — system response to target radiation 
T = detector relative response 
e = emissivity 

cr = Stefan-Boltzmann constant 
T a = atmospheric transmission 
T 0 = optics transmission 
O = solid angle 

Subscripts 1 and 0 = specific values 

18.3. Optical System 

The following functions are usually most important in optical system design gen¬ 
eralities: 

1. Sensitivity: use large relative aperture (f/ no), large aperture, high efficiency. 

2. Background discrimination: make instantaneous field equal to target subtense. 

3. Coverage: large enough to cover required area, but small to reduce scan rates. 

4. Simplicity: keep simple, light, inexpensive, etc. 

To achieve sensitivity, a field lens and reimaging technique can sometimes be used. 
A lens is placed about where the primary image is formed. It is made large enough 
to gather all the rays, and usually images the entrance pupil onto the detectors. The 
optics no longer image the target, but they uniformly irradiate the detector (see Sec. 
10.2.4). The effective /'/no of the system is usually the angle of convergence on the 
detector but sometimes includes obscuration effects. Primary optics are chosen from 
among the Schmidt, Cassegrain, Bouwers, and Newtonian telescopes. Others are 
sometimes used, and perhaps someday lenses may be used. Depending upon the appli¬ 
cation, the entrance pupil or the image plane is imaged onto the detector. 

The equations for imaging the image are: 

L = 2 F tan 0/2 



6 = tan -1 L/2o 

of 

o- f 


i - 



OPTICAL SYSTEM 


733 


where © = field angle 

6 = secondary field angle 
F = primary focal length 
f= secondary focal length 
D — primary diameter 
d = secondary diameter 

L = maximum linear dimension of image plane 
o = object distance for secondary lens 
i = image distance for secondary lens 

A typical problem can be stated as follows: for a primary objective of diameter D, 
image distance F (focal length for infinitely distant object points), and total field angle 
0, design a field lens which collects all the radiation and which refocuses the image at 
some given distance s behind the primary focus. The designer must then decide how 
big to make the lens, and what focal length and what field of view to use. 

The equations can be normalized with respect to L. The geometry is given in 
Fig. 18-1. The equations are 


d 

L 


Do 1 o 

FL + ~ T Z + 


6 = arc cot 2o/L 



A second type of problem calls for imaging the entrance pupil (usually the same as 
the aperture stop and the primary) onto the image plane by placing a lens at the primary 
image. Then, 

o = F + s 


J D 

d = — s + L 
t 

where s = distance of lens behind primary image. 

The angular field coverage is not important; it is only necessary to collect all the 
energy and "smear” it over the surface of the detector. 





















734 


SYSTEM DESIGN 


18.4. Scanning Dynamics 

The problem is to choose an optimum kind of scan pattern and to determine the 
realism of a chosen instantaneous field, total field, frame rate, detector number, etc. 
The required dwell time is 

t = coT / a 

where a> = solid angle of instantaneous field 
O = total solid angle 
t = dwell time per detector element 
T = allowable field time 

If a single detector is used, it must have a time constant r given by 

T — tig — loT/gfl 

where g = number of time constants per dwell time (usually 1 to 3). The following 
equations give prism rotation rates and instantaneous fields of view for a mapping sys¬ 
tem. (The meanings of r, 6, a, and b are made clear in Fig. 18-2.) 

s = V(27 Tgnpy'ivlh) (1 It) 

a = V(27 rn/gp)(v/h)T 
b ~ ha sec 2 6 


a = ah sec 6 


where a = instantaneous angular field of view 
p = number of detector elements 
s = prism rate 

1 In = fraction of a circle scanned 
r = time constant 
vlh = velocity-to-height ratio 
If an array of N detectors is used, then 


T = 


N a) 
g a 


T 


Usually a scanning system has less than 100% duty cycle. Several consequences 
are possible singly or in combination. 

1. Less field is covered. 

2. The field is covered less frequently. 

3. The scan is speeded. 

4. More detectors are added or other design changes are made. 

If the duty cycle is increased, the dwell time can be increased: 




where D c = fractional duty cycle. 




SCANNING TECHNIQUES 


735 


N 


\ 


\ 


\ 



The required bandwidth is related to the time of a single look at the instantaneous 
field of view. To a good approximation, 

A f= 1/2 r 

18.5. Scanning Techniques 

18.5.1. Rotating Wedges. A single wedge deviates the ray by the usual deviation 
equation for a prism. The equations are 

8 = 8i + 0 2 + oc 

ri . f 

sin 6-2 = — sin \ tt — 
n [ 


f 

n 

= — sin 
n 

In the special case of normal incidence, 


n • 

a + arc sin — sin 6 1 
n 


a + arc sin , sin | 


sin 0 2 = — sin a 
n 


where ri = prism material index 
n = environment index 
0, = incident or exit angle 
a = prism angle 
8 = deviation angle 

The refraction takes place in a single plane so that a ray is bent only in one direction. 
Assume that the direction is the x axis; then 0 2 is 6 X . As the prism is rotated about 
the 2 axis, the angular direction of the ray changes. The resultant angle can be thought 



















736 


SYSTEM DESIGN 


of as the combination of two vectors determined by the position of the prism and its 
refraction. When two prisms are used in series the deviations are combined. Two 
angular rotations are summed vectorially: 

0 = 0 , + 0 2 = | 0 ,| e j(a, > <+8 « ) + 10 2 1 e> (w * ,+6 * ) 

where co* is the rotation rate of the ith wedge 

8 i is the phase reference to the x-axis of the ith wedge 

By consideration of the relative rates of rotation and the phase angles of two equal- 
angle prisms several useful scan patterns can be generated: 

oj, = a> 2 , 8 i = 8 2 => a circle is generated as if there were a bigger prism 

ooi = —oo 2 , 8 i — 8 2 = 7 r, => a point results as if there were no prism 

001 = co 2 , 81 ± 82 = k a circle is generated with phase delay and reduced 

amplitude 

001 = ±kc o 2 , & = constant loops within a circle result; the greater the k the more 

loops 

w,# 002 , => a spiral is generated with loops decreasing as the in¬ 

equality increases 

18.5.2. Other Scanning Methods. 

Of the many types of scan pattern, the most useful ones are: 

1. Circular 

2. Palmer 

3. Spiral 

4. Rosette 

5. Hypocycloid 

6 . Raster 

The circular scan can be generated by a rotating off-axis mirror, lens, prism, etc. 
Either the primary, secondary, or a folding flat mirror or a wedge can be used. The 
radius of the circle depends upon the angle generated and the distance from the gener¬ 
ator to the image plane. The angle of ray inclination is about equal to the angle of 
inclination of a lens, usually twice the angle of inclination for a mirror and equal to the 
deviation angle for a prism or plate. (These deviation considerations apply to the other 
scans as well.) 

The Palmer scan can be generated by translation of the circular scan; geometrical 
considerations provide data on overlap. 

Spiral scans can be generated by a second motion superimposed on the rotating 
elements providing circular scan. The motion is a variable tilt angle. 

Rosettes and hypocycloids are generated by superimposing two circular scanning 
motions. Hypocycloids are generated by the superposition of two circular motions. 
If the slowly moving element has a much larger deviation, the pattern is almost a small 
sinusoidal excursion on a circle. The number of loops in the pattern depends upon the 
relative speeds of the two motions, and the depth of the "petals” depends on the relative 
angular deviations. 

Rasters are usually combinations of two orthogonal linear motions. 

A raster scan is usually to be chosen for search and imaging work (particularly when no 
prior position information is available); rosettes and hypocycloids are useful for tracking. 


EXAMPLE OF DESIGN FOR A SEARCH SYSTEM 


737 


18.6. Example of Design for a Search System 

This discussion of the design of a representative search system is divided into (a) 
requirements analysis; (6) design analysis; (c) definition of primary system parameters; 
and id) specific problems. It is an outgrowth of one author’s specific experience and as 
such represents a combination of the several system designs in which he was involved. 
The problem is to design a ballistic-missile detection system operating from an aircraft. 

18.6.1. Requirements Analysis. 

18.6.1.1. Target Radiation. The assumed target for this discussion is a solid-fueled 
missile; this can be crudely approximated by a 1000°C blackbody that provides a 
total radiant intensity of 200,000 w sr -1 . The altitude-time profile of the target is 
parabolic, reaching 300,000 ft in 100 sec. 

18.6.1.2. Problem Geometry. The range at which ballistic missile detection is 
generally desired requires consideration of earth curvature to define the limiting 
conditions of detection range. 

To find the maximum sighting range to the target at any given altitude, one must 
calculate the sum of the target distance and sensor distance to a line tangent to the 
effective horizon. The altitude of the effective horizon above the surface of the earth 
is a function of cloud cover and transmission. One must find a proper minimum horizon 
altitude h 0 for his spectral region. 

In this sample problem, detection of the missile from an aircraft at 40,000 ft with 
the horizon at 30,000 ft will be assumed. The result of the calculations described above 
is to show the length of time the target is in view before burnout as a function of target¬ 
sensor range. Such a curve would show the time available for detection as a function 
of problem geometry. This geometry-directed range is approximately 480 n mi. 

18.6.1.3. Atmospheric Transmission, Radiation, and Spatial Filtering. The trans¬ 
mission of the atmosphere is required to determine the irradiance from the target in 
the spectral region of detector response. A relatively crude calculation of transmission 
in the detector bandpass is normally adequate in defining initial system design param¬ 
eters. For purposes of this example, two spectral bands will be considered: 1.8 to 2.7 g. 
(lead sulfide) and 3 to 5 /x (indium antimonide). Net effective integrated transmission 
of the bands will be taken as 30% over each band. In a real problem, if selection of a 
wavelength band of operation is important, detailed analysis of the transmission sup¬ 
ported by repeatable missile radiation analysis is desirable. 

The radiation associated with the atmosphere and discrimination against it in 
favor of a real target is often the most significant atmospheric effect in infrared sys¬ 
tems. In the current analysis, the problem would be that of thunderheads and cloud 
tops presenting possible false targets. The most direct approach in achieving this 
discrimination is to recognize that the real target is far more intense per solid angle 
of its subtense than the background. Thus the search system’s instantaneous field 
of view is made as small as feasible. The background radiation in the field and gra¬ 
dient effects (cloud edges) are then minimized. The worst background level from sun 
reflection may be approximated by 10“ 2 w cm 2 sr -1 with a 6000°K spectrum which 
when viewed through the approximate 30% atmospheric transmission yields 3 x 10“ 3 
w cm 2 (of aperture) sr -1 . From self-emission through the 30% atmosphere, the 
irradiance at the entrance pupil is also approximately 3 x 10“ 3 w cm -2 sr -1 with a 
0°C spectrum. 

18.6.2. Design Analysis. 

18.6.2.1. Mode of Operation. The purpose of the sample search set is detection, 
not track-while-scan or surveillance. When it has provided positive identification 


738 


SYSTEM DESIGN 


of a target and sufficiently accurate angular information for other equipment to operate 
on the target, its function is complete. 

A detection is defined if a target-like signal is seen on two successive scans. The two 
detections are necessary because of the existence of nongaussian "noise” such as power- 
supply transients in a system. Such a two-detection system is better than a two-out- 
of-three system because (neglecting the effects of nongaussian noise) it can be shown 
that designs for three detections have reduced the signal-to-noise ratio by a factor of 
V2/3 (82%). The higher signal-to-noise ratio achieved in the two-detection system 
more than offsets its lower probability of detection. 

The azimuth coverage will be 360°. The range will be 90% of the geometry-limited 
range, or 433 n mi. The frame time is defined by the requirement for two full scans 
in the time the target moves from the altitude corresponding to 90% of range to burnout. 
This gives a viewing time of 6 sec and a resulting frame time of 3 sec. Selection of an 
elevation of field of view in this kind of detection system involves the problem of mini¬ 
mum range of operation of the "defense screen.” 

Now one plots the altitude of the horizon line against range from the defense aircraft 
and the missile altitude two frame times after crossing the horizon altitude line. From 
this one computes or plots the elevation angle coverage necessary to obtain detection as a 
function of minimum range operation. For purposes of this problem, a minimum range 
of 40 n mi will be assumed, and for an assumed missile velocity profile at 40 n mi, 
the elevation coverage required is 3°. 

18.6.2.2. Wavelength of Operation. Two wavelength regions 1.8 to 2.7 g and 3 to 
5 g, will at least initially be carried as parallel analyses. To a first approximation 
sensitivity of the two systems to the 1000°C blackbody target will be sufficiently close 
together so that the choice of wavelength must be based on other factors. These 
factors include: 

Background discrimination: Discrimination at 3-5 g with similar techniques is 
better than that achievable at 1.8 to 2.7 g. 

Complexity and cost: Operation at 3 to 5 g requires cooled detectors, with attendant 
problems of complexity and cost. 

Detector time constant: In order to achieve background discrimination, small instan¬ 
taneous field of view and the resultant high-speed scan is desirable. This requires 
the use of a short-time-constant detector which would be more readily available 
at 3 to 5 g. 

The trade-off is then one of cost and complexity with reduced background-discrimina¬ 
tion performance. 

18.6.2.3. Sensitivity. Effective target radiation from the 1000° C source radiating 
200,000 w sr _1 through a 30% transmission atmosphere between 1.8 and 2.7 g is: 

Sa\ = (S to tai)(% in band)(atm transmission) 

S 1 . 8 - 2.7 = (200,000)(0.25)(0.3) 


In the 3-5 -g band, 


= 15,000 w sr -1 


S 3.5 = (200,000)(0.32)(0.30) 


= 19,200 w sr -1 



EXAMPLE OF DESIGN FOR A SEARCH SYSTEM 


739 


At the maximum design detection range of 433 n mi, 

Si.8-2.7 


Hr 


. 8 - 2.7 


R 


max 


15,000 


(433) 2 (6080) 2 (144)(6.45) 
= 2.3 X 10~ 12 w cm -2 
19,200 


3 5 6.46 X 10 15 

= 3.0 x 10~ 12 w cm -2 

If background radiation in the instantaneous field of view is to be about that of the 
target, the required field of view to discriminate against sun scattering at 1.8 to 2.7 /x is 


H 


CO = 


larger 


H 


target 


Hsun reflections (sun radiation) (reflection) (% in band) 

2.3 X 10-12 
_ (3 X 10 _3 )(0.05) 

= 1.5 x 10 -8 sr 

For the 3-5-ft band, the angular field of view needed for discrimination against 
sun reflections is 


3 x 10" 12 


(O 


(3 X 10- 3 )(0.015) 
= 6.7 X 10 -8 sr 


The self-emission (0°C source, 3 x 10“ 3 w cm“ 2 sr -1 ) is less than 1 % of total in the 
3-5-/x band and so is less of a problem than sun reflection. No consideration has been 
given to the practical problem of small-detector fabrication. 

It is now possible to define the dwell time of a target in the instantaneous field of 
view consistent with optimized background discrimination as a function of the number 
of detectors in the search system, for the moment again without consideration of the 
practical problem of fabrication of miniscule detectors. 


t = 


Noj 

Cl 


T 


3(1.5 x 10-8)AT 

^(1.8-2.7) “ (27T-X3/57.3) 


Similarly, 


= (1.4 X 10~ 7 )N sec 

3(6.7 x 10-W 
^0-5) (2tt)(3/57.3) 


= (6.0 x 10~ 7 )N sec 

At this point it appears that, even for N = 100, if the practical problems of small- 
detector fabrication are solved so that spatial filtering to very small angles is achieved, 
the use of lead sulfide in the 1.8-2.7-ft band must be eliminated on the basis of the time 
constant. On the other hand lead sulfide can be considered if, instead of an individual 
detector field of view of 0.015 to 0.067 /x sr -1 (linear square of 0.1 to 0.3 mrad), the in¬ 
dividual field is 1 ft steradian or more with background rejection by other techniques. 












740 


SYSTEM DESIGN 


To define system sensitivity requirement, the signal-to-noise ratio required to provide 
a specified probability of detection consistent with a specified false alarm rate must 
be defined. 

It will be assumed that one false alarm per 10 7 sec is tolerable. A false alarm is 
defined as repetition of a false signal on two successive scans within an area of 100 
(10x10) resolution elements. It is assumed that processing in a computer will provide 
this logic. With one scan every 3 sec, in 10 7 sec, there are 3 x 10 6 scans. There are 
also essentially 3 x 10 6 successive pairs of frames (1 and 2, 2 and 3, etc.). The proba¬ 
bility of detection per successive scan pair required is then about (3 x 10 6 ) -1 , or about 
3 X 10 ~ 7 . Any single scan may be paired with the preceding or the following scan, 
so that single scan detection probability must be 

2P, XC a,, 2 = 3 x 10- 7 

Pi sea,, = 4 X 10- 4 

To define the required operating threshold relative to rms noise: the number of noise 
peaks per second is: 

h p = 0.775 A f 

also, 

/upper 1/2 1 

thus 

h p = 0.39 It 

The number of noise peaks in 100 dwell times is then 39. The probability of any 
single peak being above threshold must then be: 

4 X 10- 4 /39= 10 X lO 6 

From gaussian probability, 

P(I) < y = 1.5 e-» 2 ' 2 

the threshold y is computed as 4.9 times the rms noise. 

18.6.2.4. Probability of Detection and Required Signal-to-Noise Ratio. For purposes 
of this problem, the probability of detection will be taken as 99.8% for the two scan 
situation. The single-scan probability of detection must then be V99.8%, or 99.9%. 
With the threshold (v) set at 4.9 times rms noise, the required signal-to-noise ratio 
for 99.9% single-scan probability of detection is 8.2. To achieve this signal-to-noise 
ratio on the target signal, the noise equivalent input of the system for operation in 
the 1.8-2.7 -/x band is: 

NEW? = 2.3 x 10 -12 /8.2 

= 2.7 x 10 -13 w cm -2 

Similarly, 

• NEI 3-5 = 3 X 10~ 12 /8.2 

= 3.6 X 10~ 13 w cm -2 


18.6.3. Definition of Primary System Parameters. 

18.6.3.1. Sensitivity Parameters. It will be shown that, to a first order, the system 
may be defined by the expression 

4a 1 / 2 

nDD* € 0 VNT 


NEI = 





EXAMPLE OF DESIGN FOR A SEARCH SYSTEM 


741 


where NEI is noise equivalent input 

fl is solid field angle scanned per frame 
D is optic diameter 
D* is detector detectivity 
N is number of detectors 
T is frame time 
e 0 is optical transmission 

NEI 1 . 8 - 2.7 = 2.7 X 10~ 13 w cm -2 
The values for inclusion in the equations are 

H = 27t( 3/57.3) = 0.33 sr 
T = 3 sec 

D* = 5 X 10 10 cm (w sec)~ 1/2 

Then: 

5.2 V033 _ 

5 X 10 10 Y / 3(2.7 X 10- 13 ) 



= 130 


For 3-5 fx and with D* — 5 X 10 10 

dVn~ = 100 


18.6.3.2. Selection of Specific Values for D and N. Selection of specific D and N to 
achieve the required sensitivity involves trading off size, weight, complexity, and relia¬ 
bility, and detector array availability as well as scan-pattern generation if the full 3° 
elevation field is not covered by the array (say to meet background rejection by filtering). 

Neglecting spatial filtering for the moment and considering current detector array 
availability, a 10-in. aperture with 25 lead sulfide or 15 liquid-nitrogen-cooled indium 
antimonide detectors would seem to be a reasonable compromise. The fields of view 
and dwell time of square detectors are: 

1 . 8 - 2.7 p 


w 


T 


0 



4.3 

0.015 


= 0.014° 2 = 4.3 p sr 


(1.4 x 10 _7 )(25) = 1.0 msec 


3-5 p. 


a) 


= = 0.04° 2 = 12 At sr 


T o = 


0.12 

0.067 


(6.0 X 10~ 7 )(15) = 1.6 msec 


With the lead sulfide, the dwell time is essentially at the reasonable time-constant 
limit. The ratio of maximum or worst background power to target power in the field 
is 4.3/0.015 ~ 300. If pulse length or cloud edge slope discrimination can be used 
to decrease this ratio, the lead sulfide system may be operable. 








742 


SYSTEM DESIGN 


The most promising system from a performance standpoint is indium antimonide 
where the ratio of worst background to signal detection is 12/0.067 ~ 180. However, 
since the dwell time of 1.6 msec is far from detector time constant limited, it may be 
possible to reduce the detector width to 0.3 mrad (0.003 in. in the 10-in. /71 system) 
thereby giving a worst background signal only 15 times target radiation and offering 
better hope of pulse-length discrimination. This indium antimonide system with pulse- 
length discrimination would be the primary system. 

18.6.3.3. Other System Parameters. 

Optics. The 3° field-of-view, f/1 system requiring resolution of 0.2 mrad at 1.5° off 
axis is within current design capacity. 

System Block Diagram. For the signal-processing gimbal drive and stabilization, 
see Fig. 18-3. 

Cooling. A closed-loop liquid-nitrogen-transfer system can be developed for this 
application. 

Signal Processing. The primary signal processing system uses separate solid-state 
preamplifiers for each detector and is within the state of the art for lead sulfide or 
indium antimonide. A hard age loop referenced to a common buss is used to insure 
that the average noise level out of all detectors is the same. Saturation of the age 
signal only slightly above noise peaks (say 10 to 11 times rms noise) insures that any 
strong signals will not seriously degrade channel sensitivity. The outputs of the 



Fig. 18-3. Possible signal processing. 












































































































TRACKING SYSTEM DESIGN 


743 



Fig. 18-4. Alternate signal processing. 


detectors are commutated using diode logic matrices. A threshold set at 5.3 times 
rms noise is used at the output of the commutator. Signals above threshold are avail¬ 
able for display and computer logic use. 

Alternate signal processing is shown in Fig. 18-4, for recording only and for separate 
age and threshold before commutation. Choice of threshold before or after commuta¬ 
tion is a detail design decision. 

Because of the continuous rotation of the scanner and the resultant need for slip 
rings, location of the commutator on the gimbals would be very desirable. The pre¬ 
amplifiers should be placed close to the detectors. 

A continuous rotation-stabilized scan is to be used. This command is generated 
by a driven resolver. It is stabilized from a vertical reference system to provide azimuth 
and elevation gimbal position commands and detector roll stabilization. The scan 
will be parallel to the horizon to avoid crossing the high gradient of the horizon. Azi¬ 
muth and elevation stabilization are conventional in search systems. The detector 
roll stabilization serves in large vehicle roll angles to maintain full field coverage. 
Because of the symmetry of the optics, they need not be roll stabilized. 

Scan position pickoff will be generated from the azimuth program command. Servo 
slaving errors will be made small enough to make this an accurate indication of posi¬ 
tion. Elevation position from commutator drive will also be provided to computer 
and display. 

18.7. Tracking System Design 

In order to develop and discuss problems which are related to track system design, 
a specific problem, that of tracking aircraft from an aircraft-borne tracker will be used. 
Analogies to other combinations can be drawn by the system designer. 






























744 


SYSTEM DESIGN 


18.7.1. Conception. 

18.7.1.1. Acquisition. In the conventional system, a search set (which may be 
integral with the tracker) will have defined the target position with some accuracy. 
This acquisition field of existing systems is about 2° to 4°. In general, also, the target 
will be at a relatively long range so that problem geometry cannot include very large 
angular rates of the line of sight. The problem of acquisition, then, is generally to 
bring a quasistationary target to the boresight of the tracker in a time which is short 
compared with the total problem time. Problem time in air-to-air intercept will 
actually be measured in tens of seconds so that acquisition time is 0.33 to 0.5 sec from 
initiation of the acquisition cycle. In a linear system, this defines a servo loop band¬ 
width whose response is about 0.1 sec or a closed-loop frequency of 10 rad/sec. Whether 
a linear system is used in acquisition is a function of overall system considerations 
including such factors as whether acquisition and track use the same error-sensing 
and servo-loop logic and the specifics of error-sensing techniques. 

18.7.1.2. Track. It can be shown that in air-to-air intercept, once a target has been 
acquired, a second-order track-loop servo and stabilization against aircraft motion can 
keep track errors to less than 1/4°. Thus, the track field of view diameter may be made 
as small as 1/2°. Some advantages of using the small track field are: 

1. Increased sensitivity, allowing continued track if target radiation is reduced by 
aspect or target use of radiation suppression techniques. 

2. Background discrimination, including cases in which the target may move across 
severe backgrounds such as the horizon or bright cloud edges. The small field 
reduces the probability of severe background in the field and the intensity of 
these backgrounds when they do cross the field. 

3. Decoy discrimination, in which any flares are dropped by the target will be less 
effective because of the use of the small-field second-order servo. 

Practical difficulties have generally discouraged switching field size in existing 
systems. 

18.7.1.3. Sensitivity Analysis. A similar sensitivity analysis to that devised for 
a search system may be applied to a track system, that is, limiting optimized sensitivity 
defined by the system parameters of field of view, frame time (or servo response time), 
and number of detectors (as well by the physical parameters of optics diameter and 
detector’s Z)*). For purposes of this analysis, it will be assumed that the system is 
operating properly if quasistationary target tracking is maintained within the tracker’s 
field of view under the influence of internal random noise and drift disturbances. 

The track loop may be divided into three basic elements: an error sensor which 
generates an electrical signal as a function of target error angle; a demodulator which 
processes the signal into the form necessary to drive the gimbals, or its equivalent; 
and the gimbal drive, which generally acts as the smoothing element in the loop. The 
loop is closed from the gimbal drive to the error sensor. 

Three primary classes of error-modulation techniques have been used: (1) amplitude 
modulation, where off-axis error is a function of percentage modulation and phase 
defines polar direction; (2) frequency modulation, where error is proportional to devia¬ 
tion and polar sense to phase of the deviation; and (3) pulse position, which is a track- 
while-scan technique. 

It should be emphasized that, in a modulation error-sensing track system, the error 
information is contained in the modulation, not in the carrier. The modulation signal- 
to-noise ratio for maximum error signal as seen at the servo filter (which it is assumed 
is maintained at a constant closed-loop frequency by age or clipping as required) that 


TRACKING SYSTEM DESIGN 


745 


determines whether tracking will be maintained. The signal-to-noise ratio required 
in the servo bandwidth may be defined by recognizing that a noise pulse in this band¬ 
width equal to the full modulation error signal will, to a first order, result in motion 
of the system to remove the target from the field of view. If it is desired, then, that in 
100 time constants the probability for loss of track is to be 1%, this requires a probability 
of 10~ 4 per time constant. To find the signal-to-noise ratio corresponding to this proba¬ 
bility, reference is made to the search false alarm analysis, modified to correspond to a 
carrier system. The number of positive peaks per integration time N, is found as 

f0ll0WS: T-112* If 

N = 0.64 A/' 


NT = 1/2tt(0.64) -- 0.25 


Since either "positive” or "negative” peaks can cause loss of target, 2 NT is used, or 
0.5 peaks per servo time constant. The probability of a noise pulse to be used is thus 
2 X 10~ 4 , and the signal-to-noise ratio required in the servo loop bandwidth is approxi¬ 
mately 4.5. This ratio may then be traced back through the demodulation block to 
define required error sensing signal to noise. If the demodulation process is linear 
(synchronous), the computation is simple. The selection of a modulation frequency 
and bandwidth is not generally significant, and performance is defined by the servo 
bandwidth. For the more usual nonlinear demodulation process (AM detection, FM 
discriminator detection, etc.) the signal-to-noise ratio of the carrier system required 
to achieve 4.5 in the servo bandwidth is generally greater than that for the linear 
system and system performance is not optimized. This poor sensitivity match between 
optimized and realized systems stems from servo-loop stability considerations, e.g., the 
sampled data problem in a track-while-scan system. To reduce the servo phase lag from 
the sampling process to less than 20°, the sampling frequency must be 10 times the 
servo bandpass. For a 10° lag, a ratio of approximately 20 is required. In real systems, 
the lags from low-frequency shaping and high-frequency cuts as well as gain-stability 
variation are such that all controllable lags must be kept low. 

A track-while-scan system is usually designed with a threshold applied to detector 
output, after which error is found by pulse time or position in a demodulator. With 
this now linear (threshold) detection process, the result of the required high scan 
rate (10 times servo response) is to reduce the signal-to-noise ratio by Vlo = 3.3 com¬ 
pared with the optimized system which provides new information at the servo response 
rate. A similar level of problem exists in the other nonlinear detection systems (includ¬ 
ing AM and FM discriminator). 

18.7.1.4. Field of View, Servo, and Accuracy Analysis. It has already been indi¬ 
cated that acquisition field size is defined by search-system target-pointing accuracy 
and that the track field may usually be made quite small. The factors defining track- 
field size include: 

(1) Maximum target crossing velocity component at minimum range, 1/m, defined 
from tactical analyses and provided as inputs to the infrared system design. 

(2) Target maneuver capability, provided as inputs to the infrared system design. 

(3) Imperfect stabilization under own vehicle motion resulting from infrared sys¬ 
tem stabilization dynamics and vehicle input motion. Vehicle motions are 
provided as inputs to the infrared system design. 

(4) Biases, nonlinear friction, and other design limitations of the infrared system. 


746 


SYSTEM DESIGN 


The track loop frequency response will generally be 6-12-6-12 although 6-18-6-12 and 
others have also been used. The first breakpoint from 6 db/octave to 12 db/octave occurs 
at a frequency low enough to exceed problem time (1/an > problem time). The second 
breakpoint (12 to 6) occurs at approximately one-third crossover (0 db gain) frequency 
and the 6 db slope exists for stability considerations. The ratio of w 3 /w 2 = 10 and the 
last slope exists to eliminate high-frequency noise from the system. The ratio 013/^2 
= 10 provides for system stability over normally expected system gain variations. 

The problem of acquisition-error curve shape and servo-loop shaping to place the 
target in the track field of view will be considered under assumptions that: 

(1) Acquisition and track modes have a common front end. 

(2) Acquisition servo shaping is optimized separately from track shaping. 

The common-front-end consideration dictates the desirability (in AM or FM systems) 
of full error modulation over the smaller track field, since this optimizes track accuracy 
and background discrimination. Thus, the desired error-curve shape in such systems 
would be linear as far as the track field, with full modulation at maximum expected 
track error, and then flat as far as the edge of the acquisition field. Background 
discrimination in track is aided by the nonlinear error-curve shape (compared with a 
linear error curve) since elements outside the track field contribute no more modulation 
than those within it. 

In the pulse-position system, the common acquisition and track fields present no 
problem since a linear error curve may be indefinitely extended for acquisition, with 
no loss in track accuracy. Track-field sensitivity does, however, decrease with such 
an increased field. During track, background discrimination in the pulse-position 
system may be achieved by logical rejection of any error signal outside the track field. 

If acquisition servo-loop shaping is to be optimized independently of track shaping, 
the acquisition closed-loop frequency is chosen by trading off sensitivity and acquisition 
time, utilizing maximum tolerable acquisition time to maximize sensitivity. The effect 
(on acquisition time) of the saturated error curve in AM and FM modulation systems 
is to increase acquisition time for a fixed closed-loop design frequency (assuming this 
design is based on the linear portion of the error curve). In addition, if the system 
must remain in the acquisition mode when it is within the track field, servo stability 
may be adversely affected unless a simple first-order design is used, since a 6-12-6-12 
loop design will cause excessive overshoot at the reduced gain resulting from the 
saturated error curve. If logical switching to track is accomplished, the tendency 
to overshoot may, however, be successfully used to shorten acquisition time, provided 
proper damping of initial angular rate occurs when the switch to track is achieved. 

18.7.2. Sample System. 

(A) System Input and Limitations. The parameters around which the sample 
system are to be defined are: 

(1) Target Radiation. 1000 effective watts per steradian in the detector wave¬ 
length band (3 to 5 p.). 

(2) Target Motion. Maximum requirements of servo response will be based on a 
target at 1 n mi range with a crossing velocity of 1000 fps and a 4 -g target turn at 1 n mi. 

(3) Acquisition 

(а) Field of view: 4° diameter 

(б) Time: 0.3 sec 

(4) Track Accuracy. 2 mrad under worst conditions 

(5) Space. Allowable front-end cross section: 10 in. diameter 


TRACKING SYSTEM DESIGN 


747 


( B ) Front-End Description 

(1) Optics. Allowing for covering dome, clearance, and gimbals, an optics diam¬ 
eter of 7.5 in. may reasonably be achieved. An /VI refractive or folded catadroptric 
system is required for short length to allow for maximum gimbal angle coverage. For 
background discrimination, optical resolution of 1 mrad or less to 2° off-axis is re¬ 
quired. A bandpass filter in the general range of 3 to 5 /x is required. Expected 
net optical transmission over the band is about 35%. 

(2) Detector. A cruciform detector array canted 45° to the horizon is used, 
subtending 4.5° along each axis by 1 mrad of detector width. The 45° angular tilt 
results in a scan which will not sharply cross the horizon, the primary background 
difficulty. Detector material is indium antimonide with a peak D* in the band of 
5 x 10 10 cm cps 1/2 w -1 . To avoid excessive system complexity, the same detector 
and scanning should be used in acquisition and in track mode. 

(3) Scanning. Image nutation over a 4° field is caused by rotation of an optical 
wedge or a tilted mirror. 

(4) Track-Loop Servo Analysis. Servo requirements of the track loop are de¬ 
fined, and the adequacy of acquisition time is evaluated. To be consistent with a worst 
condition of 2-mrad error, linear dynamic lag is maintained at 1 mrad. The maximum 
angular rate may be calculated as 9.6° sec -1 , and maximum angular acceleration is ap¬ 
proximately 1° sec -2 . Angular acceleration from a 4 -g target turn at 1 n mi is 1.2° sec -2 . 
The effective velocity constant must then be 170 sec -1 at the "time” of maximum angular 
rate, and the acceleration constant must be 21 sec -2 . Frequency response of the 6-12-6- 
12 loop to achieve these K v and K 0 values will result in an 8 rad sec -1 crossover fre¬ 
quency. The simplest acquisition computation to make is for a first-order system. 
The 6-12-6-12 system described here will have a faster response. For a target 2° off-axis 
at acquisition and an 8 rad/sec first-order loop, the time to reduce error to 2 mrad, or 
0.059 of the initial value is approximately 0.3 sec; or to state this another way, specifica¬ 
tion reduction of acquisition error to less than 1 mrad requires approximately 0.4 sec. 

With the 8 rad/sec cutoff frequency, the required information rate to achieve less than 
10° phase lag at the servo closed-loop frequency corresponds to an angular frequency 
of about (20)(8) = 160 rad/sec, or 160/2 tt = 25 pps. Using a pulse-position technique, 
with a small track field, two independent error pulses per scan are obtained. Thus, 
a scan rate of about 13 cps should be used. This scan rate is a significant parameter 
in defining the sensitivity of a pulse-position error-generation system. 

(5) Sensitivity and Range Calculation. Having defined the significant front- 
end parameters, it is now possible to compute system sensitivity and range performance 
on the specified target. The expression for noise equivalent intensity (NEI) is: 


NEI 


NEP r 

{A„)(e 0 )r 


where NEP r is the noise equivalent power on the detector, A 0 is the optical collecting 
area, e„ is net optical efficiency, and r is the system response to the input signal. 

NEP r = VA C A f/D* 

where A c is cell area and A/'is bandwidth, from which we can calculate 

VA c bf 

D* ( A 0 )(e 0 )r 


NEI = 






748 


SYSTEM DESIGN 


Optimum bandwidth in the single-pulse system, assuming a sinusoidal pulse, is 

1.84 0.3 

' (277 -)T T 

With a 4° circle scanned at 13 cps and a 1-mrad detector width, 

(0.001X57.3) 


T = 


(7t)(4)(13) 


Thus, 


A f = 


0.3 


0.00035 


= 0.00035 sec 


— 860 cps 


Low Pass: 
able as: 


Cutoff in the present example is 100 to 200 cps. 

r = 0.67 


Response factor is deriv¬ 


ed/ Area: Using a single cell per arm, 

A c = Clf 2 

where Cl = solid angle and f = focal length. If 

Cl = (0.001X4.5/5.3) = 7.85 x 10- 5 sr 
and the focal length is 7.5 in., 

A c = (7.85 x 10- 5 )(7.5) 2 (6.45) = 0.00285 cm 2 
Aperture area has been given as 


A npertur e = ^ (7.5) 2 (6.45) = 285 cm 2 


Optical efficiency has been given as: 


and 

Thus, 


e 0 = 0.35 
D* = 5 x 10 10 


NEI 


V(0. 0285X860) 

(5 x 10 10 )(285)(0.35)(0.67) 


1.5 x 10 -12 w/cm 2 


Signal-to-Noise Ratio for Acquisition: A tolerable false alarm rate of 1 in 10 servo 
response times (once in 10/8 = 1.25 sec is assumed. Since acquisition does not require 
a high "hit” percentage of signal, 50% probability of detection is assumed. 

With the 860-cps bandwidth, the number of signal peaks in 1.25 sec is: 


IV,. 25 = (1.25X0.775X860) = 800 


False alarm probability is then 1/800, or 0.0013. A threshold is then required at 
4 times the rms noise. The signal-to-noise ratio for 50% probability of detection is 
also 4. Thus, a target which is to be acquired has a radiation at the system aperture of: 

7 = 4 NEI 

= 6 x 10 _ 12 w/cm 2 


Detectable Range Target: Assuming 30% atmospheric transmission, a 1-kw/sr target 
may be detected at: 








TRACKING SYSTEM DESIGN 


749 


r = 



(1000X0.3) 


6 x 10- 12 


= 7.1 x 10 6 cm 


= 38 n mi 


Comparison of Sensitivity with a Servo Bandwidth-Limited System : The field in a 
servo bandwidth-limited system is effectively scanned once per servo response time, 
but a signal-to-noise ratio of about 6 is required. The system described here is thus 
compared with the servo bandwidth-limited system by the ratio: 

sensitivity-instrumented system _ 6 
servo bandwidth-limited system 4 

A servo bandwidth-limited system would thus have twice the sensitivity of the in¬ 
strumented system. 

(C) Block Diagram. The block diagram of the sample system is shown in Fig. 18-5. 
By tracing through the significant elements of this diagram, the trade-off logic is 
demonstrated. 


8 


2tt(13) 


= 0.48 



Fig. 18-5. System block diagram for track. 



















































































































750 


SYSTEM DESIGN 


(1) Error Generation. Cruciform-detector outputs will be separately amplified 
in gain-stabilized amplifiers with strong signal limiting to avoid amplifier blocking 
by saturation. Amplifier outputs drive threshold circuits to remove noise from the 
signal at a level defined by a tolerable false alarm rate. The threshold outputs drive 
error demodulators which define target error position. These error demodulators derive 
scan-position information from a reference generator located at the nutation drive in the 
optical system. System logic provides (in the acquisition track switching) that, after 
initial acquisition (as defined by error dropping below a critical threshold), the threshold 
circuit will not pass signals corresponding to errors greater than those expected in 
track (say, 1/4 degree). 

(2) Target Presence Generation. When enable logic initiates acquisition, an 
interlock based on detection of a target in both detector channels is provided. When the 
switch to small-field track mode is initiated, the target-presence circuitry operates only 
on track field signals. Loss of signal for any reason should probably be used to initiate 
a return to the search mode. 

(3) Stabilization of Track. Two stabilization components are used in this sam¬ 
ple system. The less conventional one is roll stabilization of detectors for background 
discrimination to avoid severe horizon crossing, which may occur in maneuvers even 
with the 45° rotated detectors. This roll stabilization need be only approximate. It 
receives its command from aircraft instruments. The effect of detector roll position 
must be considered in error generation (reference generator affected) and in coordinate 
rotation of the error signals necessary to drive the conventional azimuth-elevation 
gimbals. 

The more conventional stabilization is that about the line of sight to the target to 
maintain track with respect to the tracking aircraft’s maneuvers. Before one considers 
the analytical parameters of this stabilization, the problem of placement of stabilization 
gyros should be settled. Conventionally, they are placed on the sighting platform, and 
if space and heat-sink problems permit, every effort should be made to locate the gyros 
on the IR telescope. If this is not possible, a separate stabilization platform may be 
provided to which the sighting head is slaved with an aided rate command. This tech¬ 
nique has been used successfully on at least one infrared equipment. Sight line re¬ 
mains to the target since the position loop provides at least one integration to remove 
any slaving errors. In Fig. 18-5, stabilization is shown using an auxiliary platform. 

Analytical treatment is independent of the gyro location. An electric-motor-driven 
geared gimbal system is adequate for this class of equipment. With conventional gear 
trains, stabilization closed-loop frequency is generally limited to 8 to 15 cps. In this 
treatment, a 65 rad/sec ( — 10 cps) loop is assumed. The basic loop response is shown 
in Fig. 18-6. The "long-period” velocity constant is 980 sec -1 , and acceleration constant 
is 1950 sec -2 . The stabilization loop need not be a high-order loop below frequencies 
at which the track loop is tightly closed. The acceleration constant of 1950 sec -2 
means that a 1-mrad dynamic error may be maintained under acceleration inputs of 
approximately 110°/sec 2 as seen at the gimbals. 

(4) Cooling. Depending upon mission time, logistics, and weight trade-off 
penalities, an open-loop or closed-loop cooling system decision may be made. 

18.8. Mapping Systems 

18.8.1. Requirements. This section does not deal with the problem of extracting 
the information from a map (often the critical element in mapping-system utility) but 
only with maximizing map-information content, which is achieved by use of a small 
resolution-area detection of minimum contrast combined with scan of a large field. 
As usual, reconciling these quantities involves compromise. 


MAPPING SYSTEMS 


751 


\ 



18.8.2. Basic Sensitivity Equation. 

18.8.2.1. Detector-Noise-Limited System. The derivation of the sensitivity equa¬ 
tion for a detector-noise-limited system may always be started from the simple detector- 
sensitivity expression: 

NEP C = VAc Aj/D* 

and the resultant overall system expression: 

_ NEP C VA c Af FD /0.65H F [a 

(A 0 )(e 0 )r D*A 0 e 0 r D*(7rD*/4)e 0 r V T f 1,1 D*De 0 r V T f 

The detectable signal may be found by computing the incremental signal at the 
aperture and setting it equal to the NEI multiplied by some constant. This constant 
for detection in a single resolution element would have to be same as for a point detec¬ 
tion system (about 4 to 7 times the NEI). However, with the element-to-element 
integration of a map, larger-area information is derivable at a lower signal-to-noise 
ratio down to below unity in a single element. 

The incremental radiation dW may be found from: 

W = eaT 4 


dW = ctT 4 de + 4 earT 3 dT 
= W M dt +4eW M Y~ 

where e = emissivity 

<t = Stefan-Boltzmann constant 

Wbb = blackbody radiant emittance (e = 1) 

Normal system use is limited by detector response and atmospheric transmission 
to a spectral band less than that essentially encompassing total radiation. Also, 
atmospheric transmission may further attenuate the signal. The incremental radia¬ 
tion dH at the aperture in a resolution-element solid angle co is then: 

/ dT\ to 

dH = [W^^de + 4eW Xl .x 2 —J t a ~ w cm 2 


















752 


SYSTEM DESIGN 


where VTx,-x 2 is total blackbody radiation (over a hemisphere) per unit area in the in¬ 
terval from Xi to X 2 , t a is atmospheric transmission in the Xi-X 2 band, and o/tt is the 
ratio of radiation in the instantaneous field of view, co, to the full radiation of the body 
over a hemisphere. 

For a given background environment, it is seen that, within practical limitations 
such as detector response time, the product ode or o dT permits a trade-off between 
resolution and minimum detectable temperature difference. 

18.8.2.2. Background-Radiation-Noise-Limited System. Some infrared detectors 
are background-radiation-noise-limited when they view an ambient temperature 
background. A design for this type of detector is derived in this section. 

It is assumed that the detector has cold shielding, allowing a solid angle of radiation 
acceptance, 8, which is twice that corresponding to the geometric f number, J ; that is 

8 = 2 / 3 2 


It will also be assumed that temperatures inside and outside the instrument are the 
same and that by reflection of similar temperature sources the net background within 
the angle o may be approximated as a blackbody. 

If the instantaneous and total fields of view of the system are o and O and the frame 
time is T f , detector area A c is related to o, 3, and D 0 by: 


The dwell time on target is: 


Again using a bandwidth 


A c = (oS 2 D 2 


t = 


O Tf 

~n~ 


A f = 


0.65 

t 


0.650 

(oTf 


The rms noise due to radiation noise in a bandpass A f is the same as that in an inte¬ 
gration time 


1 oTf 

2lf = 1.30 


The background radiation in the sensitive band Xi to X 2 received by the detector 
through a solid angle 8 is 


H'=W r x l .x ! A c ^ = H\,.kXf ! 0 ! 


2 
7 T 


= — Wx.-kooD 2 


If the detector has a quantum efficiency Q, the number of effective background photons 
received in the integration time, T„ is: 


N b = Q 


W 


hc/\ ( 


Ti 


Q - Wx^oD 2 

7T (til f 


= 0.49 


hc/\ c 1.30 

Q\ r Wx l .x 2 (*> 2 D 2 Tf 

hcCl 












MAPPING SYSTEMS 753 

where X, = center wavelength of Xi-X 2 and he = the number of noise photons in the 
integration time, 7\, is: 


Ns — V Nn — coD 


yj (0.49) 


QK c W kl . Kt T f 

hcCl 


The detector NEP is: 


Then: 


NEP, 


Nvti he 
T i QX C 


l-3f IcoD [~TQKWZ^Tfhc 

QcoT f V * hen X, 


NEP, = 0.91Z) \ — 


he Cl 


X, QTf 


W 


x i -X 2 


NEI 


NEP, NEP, 
A 0 e 0 r 7 tD 2 


with r = 0.9 


0.91 D J— 


a 

QTj 


W 


x,-x 2 


ttD 2 


e 0 r 


l h±JL w 


= (1.3) 


X, QT J 


X i -X 2 


/ 


De n 


18.8.2.3. Multiple Detector System. If a multiplicity of detectors is used, the total 
field searched per frame time is divided by Nn, the number of detectors, and computed 
sensitivity is improved by V^Vo, or 


NEI oo—— 


It must be noted that if multiple detectors are used in a mapping system (and they 
may be desirable for increasing resolution and decreasing the required detector response 
time), and if a striped map is to be avoided, detector responsivities as seen at the output 
of the signal processing circuitry must be matched. 

18.8.2.4. Edge Resolution vs. Signal Strength. The term instantaneous resolution 
is related to the minimum detectable signal level and is limited to the resolution area 
of the detector. In the presence of a strong signal, however, full response need not 
be achieved to define an edge. To a first order then, if data-reduction techniques 
are used to best advantage, the usable resolution solid angle varies inversely as the 
contrast signal-to-noise ratio so that a doubling of signal-to-noise ratio contains in¬ 
formation allowing half the resolution of an isolated edge. 

18.8.3. Background-Limited D*-Q Trade-Off. To find the D*-Q relationship 
defining the value of detector D* that is equivalent to a specific value of Q in the system 
configuration defined in Section 18.8.2, the two NEI expressions are equated. 

























754 


SYSTEM DESIGN 


I he _n_ 

1.1 F la V X c QTf 12 
D*De 0 \T f 16 De 0 

With F = 1, W Kl -x t = (0.031X0.4) = 0.012, and \ c = 10 ^ (8- 15-/x band at 0°C), 

D* 

—= = 5.4 X 10 10 

Vq 


18.8.4. Sample System. 

18.8.4.1. Specific Requirements. The basic requirement may be stated simply as 
defining a low-altitude mapping system to operate at 1000 feet and at a velocity of 
800 fps and to provide a thermal map, in the 8-15-p, atmospheric window, of 0.5 mrad 
resolution (6 in. at a nominal 1000-ft altitude). The desired signal detectivity in the 
extended area (unity signal-to-noise ratio) is 1°C against a nominal 0°C background 
of 0.5 emissivity. The system should be capable of operation up to 10,000 ft at 800 fps 
with constant optical resolution and the stabilization required for production of good 
map information. Stabilization and image motion compensation must provide for 
10% image smearing line-to-line. 

18.8.4.2. Sensitivity Computation. As the basis for a realizable system, a D* 
corresponding to a quantum efficiency of 0.1 will be assumed. Neglect time constant 
for the moment; the optics diameter necessary to achieve a signal-to-noise ratio of 
unity for a 1°C change in temperature is found by equating the signal to the NEI. 

l.iff [a _ . ... cLT o> 

D*De 0 V T 6 x, ' x * T Ti 77 

The values for the parameters are: 

2 = 1 

D* = 1.7 x 10 10 


€o = 0.3 

a = (0.5 X 10 3 )(27r), assuming for simplicity a continuous single element with 
~50% dead time above the horizon scan 


T is found from v/h and the detector dimension along the flight line: 


T = 


0.5 x IQ- 3 
800/1000 


= 0.625 x 10 3 sec 


€ = 0.5 

Wx,-x 2 for 0°C and 8 to 15 p = (0.031X0.4) = 0.0124 w cm 2 

dT ^ 1 
T ~ 273 

t a = 0.6 

w = (0.5 x 10~ 3 ) 2 = 0.25 x 10- e 
Solving for Z), 

D = 112 cm = 44 in. 

This is certainly an unreasonable dimension. 











MAPPING SYSTEMS 755 

18.8.4.3. Use of Multiple-Detector Array. The aperture is large; the dwell time is 
short. Thus, 


— r - 0 25 X 1Q~ 6 

O 1 ~ (2tt)(0.5 X lO" 3 ) 


(0.625 X lO 3 ) 


= 5 x 10~ 8 sec 


From both a sensitivity and a time-constant standpoint, it is desirable to use a detector 
array. 

If a 10-in. aperture is to be used, 20 detectors are required and dwell time on target 
is 10 6 sec. 


18.8.4.4. System Configuration. (1) Front End. The optical system and detector 
array are mounted on a roll-pitch stabilized platform. Scan is by a 45° tilted mirror 
continuously rotating about an axis along the aircraft heading and at a speed defined 
by vlh input, discussed below. The nominal operating rotation speed will be 80 rps 
(800 fps at 1000 ft and 20 x 0.5 mrad = 10 mrad per scan). To keep the image properly 
placed with respect to the detector, a dove prism which rotates an image at twice its 
own angular rate is used. 

Detector cooling may or may not be on the stable platform depending on the trade¬ 
off between liquid-transfer problems and platform size and weight. The detector array 
is packaged in a single dewar and aligned to sight along the flight path. The detectors 
may be staggered in the focal plane rather than in a single line to avoid the problem 
of spacing between detectors. A cold shield is used to isolate the spectral interval 
and define the field of view. 

(2) Signal Processing. The main problem in the signal processing is that of 
maintaining detector-to-detector consistency. To do this, operation must be below 
detector cutoff frequency so that detector time constant is uniquely defined for all ele¬ 
ments. Amplifier frequency responses must be closely matched using stabilized circuit 
techniques. The separate channel outputs may either be separately recorded by multi¬ 
channel means or may be commutated by high-speed gates sampling 3 times per dwell 
time. Because of the resolution degradation as scan departs from nadir view a pro¬ 
grammed reduction in the number and choice of detectors for recording may be used. 

(3) Stabilization Requirements. Long-time stabilization must be better than 
2 mrad and short-term scan-to-scan stabilization about 0.1 of the system resolution, 
or 0.05 mrad. This means 0.05 mrad in 1/80 sec, or 4 mrad sec 1 rate error. At 
10,000 ft altitude, 0.4 mrad sec -1 is required. To achieve good short-period stabiliza¬ 
tion, a gearless torqued gimbal structure with self-contained gyros will be used. Long- 
period vertical reference information may be transmitted from aircraft inertial refer¬ 
ence equipment. 

(4) Image Motion Compensation 

(а) Requirement. IMC in a high-speed scanning system is a less critical problem 
than in a photographic system. With a single detector, 10% scan-to-scan resolution 
smearing corresponds to 10% accuracy in vlh. With 20 detectors, smearing of 10% on 
one detector image means 0.5% vlh accuracy. Drift-angle accuracy follows the same 
pattern. With a single detector, 10% drift corresponds to an angle of approximately 6°. 
With 20 detectors, the requirement is 0.3° to achieve 10% scan to scan misalignment. 

(б) Implementation, vlh compensation will be made a part of the system by varying 
scan rate to achieve proper matching of line to line scanning. Drift compensation 
will be made a part of the system by skewing data output. 



756 


SYSTEM DESIGN 


(c) v/h and Drift Measurement 

1. External Source: If the aircraft has a radar altimeter and doppler, inertial, 
or doppler-inertial navigation, v/h and drift angle to the required accuracy may be 
derived from these sources. 

2. Passive v/h and Drift : The mapping system itself, with some additional focal- 
plane detectors and additional processing, offers a self-contained source of v/h and 
drift information, v/h is computed by correlating data sampled from the central angle 
of the scan (directly below the aircraft) for overlap or underlap on two successive scans, 
using v/h "overlap” detectors added for this purpose. Drift angle is computed by 
correlation techniques looking for scan-to-scan skewness in data sampled from directly 
below the aircraft on detectors separated by a large angle along the flight path and 
caused to overlap by v/h to cover the same area on successive scans. The v/h and drift 
error signals may thus be generated to vary scan speed and drift angle. 


Chapter 19 

INFRARED MEASURING 
INSTRUMENTS* 


CONTENTS 


19.1. Radiometers. 

19.1.1. Reference Radiation Level. 

19.1.2. Commercially Available Radiometers. 

19.1.3. Normalization Methods. 

19.2. Spectroradiometers. 

19.3. Monochromators. 

19.4. Spectrometers and Spectrophotometers. 

19.4.1. Infrared Grating Spectrometry. 

19.4.2. Commercially Available Spectrometers 

and Spectrophotometers. 

19.5. Interferometers. 

19.5.1. Rayleigh Interferometer. 

19.5.2. Michelson Interferometer. 

19.5.3. Twyman-Green Interferometer. 

19.5.4. Fabry-Perot Interferometer. 

19.5.5. Spherical Fabry-Perot Interferometer. 

19.5.6. Lummer-Gehrcke Plate. 

19.5.7. Spectral Transmittance of Interferometers. 

*Material prepared by the technical writing staff of McGraw-Hill, Inc. 


758 

759 
761 
761 

763 

764 
766 

766 

767 
776 

776 

777 

777 

778 

779 

779 

780 


757 





















19. Infrared Measuring Instruments 


19.1. Radiometers 

A radiometer is a radiation-measuring instrument having substantially equal re¬ 
sponse to a relatively wide band of wavelengths in the infrared region. Radiometers 
measure the difference between the source radiation incident on the radiometer detector 
and a radiant energy reference level. 



Fig. 19-1. Basic radiometer. 


The basic design of a simple radiometer is shown in Fig. 19-1, where 
L = collecting optics, which forms the circular aperture stop of area A 
D = dectector element, which forms a circular field stop of area a 

0 o = half-angle, measured in radians (Cl = 2n (1 — cos 0 O ) ~ n do 2 = al f 2 ); Cl is the 
solid angle, is steradians, of the corresponding conical field 

f = focal length of radiometer 

All radiometers and radiometric measuring instruments contain at least the following 
three essential components: 

(a) A detector element, which converts changes in incident electromagnetic radiation 
into variations of an easily measured property, usually an electrical signal. 

(b) An optical system, which determines the combination of receiving aperture and 
angular field of view through which radiation is collected, thus delineating the 
amount of radiation to which the radiometer responds. The optical system 
includes the sensitive surface of the detector. 

(c) An amplifier and output indicator, usually electronic, to transform the output 
of the detector element into the desired form of presentation. 


758 


















RADIOMETERS 


759 


19.1.1. Reference Radiation Level. Both absolute and relative infrared radiation 
levels may be obtained with a radiometer. Since absolute levels of radiation are defined 
with respect to absolute zero, the level of radiant power incident upon the detector must 
be compared with a known reference level to derive the absolute radiant power level. 

The ultimate accuracy of a radiometer is determined by the form of reference radiation 
level used. The detector itself (dc radiometer) or a radiation chopping system (ac 
radiometer) can be used to provide the reference radiation level. For absolute measure¬ 
ments, the chopping system is preferred. 

Radiometers that measure the difference in radiation from two neighboring spatial 
positions provide relative information only as no reference level exists. Radiometers 
used to compare any element of a large area to the average radiation associated with the 
entire area can supply an absolute measurement, providing the average intensity is 
known and used as a reference level. Similarly, in the time domain, the power level 
due to radiation at one instant may be compared to that of a previous instant or to an 
average associated with all past measurements. 

The characteristics of three different type radiometers, including the form of reference 
level they employ, are listed in Table 19-1. 


Table 19-1. Principal Characteristics of Different Type Radiometers [51 


Type 

Detector 

Detector Response 
Speed (Time 
Constant) 

System 

Speed 

Reference 

Radiation 

Level 

Radiation 

Measurements 

Electrical 

Signal 

DC 

Thermopile 

2 sec 

2 sec 

Emissivity 
and tem¬ 
perature 
of thermo¬ 
pile 

Difference of 
source radi¬ 
ation and 
thermopile 
radiation 

Electromotive 
force from 
compensated dc 
thermopile 

AC 

Blackened 

Chopper 

Thermistor 

1 msec 

25 msec 

Emissivity 
and tem¬ 
perature 
of black¬ 
ened chop¬ 
per 

Difference 
of source 
radiation 
and black¬ 
ened chop¬ 
per radi¬ 
ation 

AC signal from 
compensated 
thermistor 
bolometer 
bridge 

AC 

Chopper 

Mirror 

Thermistor 

1 msec 

Adjustable, 
16 msec to 

1.6 sec 

Temperature 
of black- 
body refer¬ 
ence (emis¬ 
sivity = 

1.0); tempera¬ 
ture con¬ 
trolled or 
monitored 
within 

0.2° 

Difference of 
source radi¬ 
ation and 
reference 
blackbody 
radiation; 
null method 
may be used 
with tempera¬ 
ture-con¬ 
trolled black- 
body 

AC signal from 
compensated 
thermistor 
bolometer 
bridge; null 
detection 
method when 
using controlled 
blackbody 


19.1.1.1. Detector Energy Level Used as Reference Radiation {DC Radiometer). 
A radiometer that uses the detector as a radiation reference level is usually referred to 
as dc radiometer. This terminology is applicable because the instrument measures a 
change in the dc electrical properties of a thermoelectric or bolometric infrared detector. 

DC radiometers are subject to drift because the reference level is determined by the 
temperature of the detector. Drift can often be tolerated when the temperatuie of the 
target is much higher than the detector’s ambient temperature. Significant errors can 
result, however, when the target temperature approaches ambient temperature. The 


760 


INFRARED MEASURING INSTRUMENTS 


typical response speed (time constant) for instruments using thermopile detectors is 
approximately 2 — 4 sec where the response speed is time required for the instrument 
to reach He of the final response. 

19.1.1.2. Chopper Used as Reference Radiation Level (AC Radiometer). Chopper 
or ac radiometers utilize an ac output from the detector for signal processing. They 
are particularly suitable for absolute radiation measurement and do not have the drift 
problems associated with dc radiometers. The electrical output of this type radiometer 
is proportional to the difference between radiation falling upon the detector from the 
source within its field of view and that of a blackened chopper blade or a controlled 
reference blackbody. In the latter case, a chopper mirror alternately directs radiation 
from the source and the reference blackbody onto the detector. 

Blackened Chopper. Figure 19-2 shows a blackened chopper where the detector 
alternately sees the source image and the blackened chopper. The temperature and 
emissivity of the blackened chopper determine the reference radiation level. At wave¬ 
lengths out to about 1 p, a very stable reference level is obtained by ensuring that the 
chopper has a uniformly coated black surface. Even out to about 3 p, chopper temper¬ 
ature is not usually a critical factor because at ambient temperatures there is little 
radiation in this region from a blackbody or a graybody. At wavelengths longer than 
about 3 p, however, the effects of variations in chopper temperature and emissivity 
become a serious consideration. 



Fig. 19-2. Blackened chopper schematic diagram. 


Ref. Temp. °C 



Ref. 

Blackbody 

Radiometer 

Head 


|— Incoming Radiation 

Output 


Radiation Output 


Display 

or 

Recorder 


Fig. 19-3. Chopper mirror block diagram. 


Chopper Mirror. Emissivity effects can be minimized by using a highly polished 
chopper blade. However, incident radiation reflected from the surrounding area to 
the detector by the polished chopper must be controlled. This is done in the radiometer 
shown in Fig. 19-3, where reflected incident radiation is controlled by a reference black¬ 
body and a chopper mirror. 










































RADIOMETERS 


761 


A small reference blackbody is placed in such a position that, by specular reflection 
from the chopper mirror, radiation from the reference blackbody and the source image 
are alternately directed onto the detector. This not only provides a stable source of 
reference radiation but also an adjustable one when temperature controls are provided 
for changing the reference blackbody temperature. In addition, if the reference black¬ 
body source is adjustable, its radiance can be matched to the source radiance and the 
detector used as a sensitive null or quantitative error detector. 

19.1.2. Commercially Available Radiometers. Table 19-2 lists some of the charac¬ 
teristics of most commercially available radiometers. 

19.1.3. Normalization Methods. By careful design, some instruments achieve 
quite uniform responsivity over wide but finite spectral regions. In many cases, 
however, responsivity is not uniform even within the band, due to the combined effects 
of the spectral characteristics of the components (mirrors, lenses, filters, prisms, trans¬ 
ducers, etc). Also, the source spectral distribution is almost never constant. In 
general, the output signal produced by an incident beam of radiant power 

P = Jp x d\ w (19-1) 

is given by 

V=fp^R p d\ v (19-2) 


where R p = Rp (X) is the spectral power responsivity in volts per watt. Even where 
R p (X) is completely known from calibration measurements, it is not possible to solve 
Eq. (19-2) uniquely for P x (X). Consequently, it is also not possible to determine P = 
f Pk d\. However, if the relative spectral distribution p x (X) in the incident beam is 
known, the measurement will establish the scale factor P s , which satisfies 

P\ (X) = P s px. (X) w (19-3) 

from the relation 

P g = v/jp x R p dk w (19-4) 

On the other hand, without any information about the spectral distribution of the 
incident beam, the output of the instrument can be compared meaningfully only with 
measurements made with instruments having exactly the same relative spectral 
responsivity characteristic. 

Fortunately, it usually happens that there is at least some information about the 
spectral distribution p x (X), although it may be only approximate and often is only 
implicit and not clearly recognized. For example, from the known physical character¬ 
istics of a target —its material, its approximate temperature, etc —it is often possible 
to judge the general shape of the curve of p x (X) and hence to be able to estimate its 
relation to the spectral curves of other targets in order to interpret comparisons between 
measurements of two such targets by spectrally selective radiometers. In order to 
eliminate differences in other instrumental characteristics (which also affect respon- 
sivities) for the purpose of making such comparisons, normalization methods are often 
employed in reducing the data [1,2]. 

Note that, although this discussion is all in terms of radiant power P and power 
responsivity R p> all of it applies equally well to the other radiometric quantities for an 
incident beam, H and N, and the corresponding responsivities, and to the source quan¬ 
tities J and W, derived from measured values of incident P, H, or N, as well. The 
latter, however, involve additional uncertainty because of the problem of evaluating 
the attenuation between source and instrument. 


Table 19-2. Characteristics of Commercially Available Radiometers 


762 


INFRARED MEASURING INSTRUMENTS 


S' 

3 q 

o 

o 


CM 


I I 


e 

-2 - 
*♦«* CO 

o ^ 

CO S 


CM 

o 


3 


o ^ 

O'*** © 

& s o | 


iq 

CM 


05 05 


CM 


ia 

CM 


CO 


o 


8 sp s 

fe, S 3 
-a 


o 

CM 


o 

CO 


o 

CO 


to 

m 


o 

CM 


o 

CO 


o oo 
s» 3 
5® o 
e r o 
O ft. 

as 


8 

o 


8 

© 


8 

I 

o 


o 

CM 


8 

I 

in 


m 

CM 


I I 


"0 


<*> Co 

^ £ 


o 

CM 

r-H 

X 

o 

CM 


o 

o 


iq 

rH 

lO 

oo 

00 

00 

CM 

ID 

00 

X 

X 

X 

X 

X 

1 x 

O 

o 

00 

ID 

r-H 

ID 

ID 

00 

00 

oo 


CM 




o 

CO 


iD 

00 


o 

CM 


O 


co 


In. 

C 



.O 

r© 

s 

•♦o 

£ 

43 
• »•» 

O 





00 


CD 


CD 


lO 


co 

.© 

o 


g 

’3 

be 

0> 

CO 

CO 

CO 

u 


CO 

be 

0) 

CO 

CO 

cd 

O 


CO 

G 
be 
a I 

CO 

CO 

cd 

O 


T3 

C 

,P 


g 

’3 

bo 

0) 

CO 

co 

cO 

O 


cd 

u 

bo 

3 

a> 

co 

cd 

O 


ic 


C*H 

© 

o 


*5 © 

o ©, 

CH 


gOo 

PQ 


PQ 

Eh 


PQ 

Eh 


PQ 

H 


PQ 

Eh 


CO 

X) 

Ok 


PQ 

Eh 


PQ 

Eh 


PQ 


£! 

3 

-*»> 

g 

Cu 

s, 

s 

Eh 






CO 
















00 

CO 

2 



£ 

5 

co 

co 

1 

1 fi 1 1 

a 


00 

a 

o 

1 CD 

CD 

0) 

1 1 0) 


o 

a; 


co 

1 

CO 

1 

X 


CO 

-+-> 



co 

CO 

a 

E 

[> 

ID 



CD 

CD 

CO 

ID 



CM 

CM 

< 

< 

+1 

CM 




o 

o 

o 

o 

o 

p 

<D 

o 

o 

o 

o 

o 


2p ^ 

o 

o 

o 

o 

o 

CJ 

S a. 

rH 

rH 

t—H 

r-H 

rH 

5, 

a w 

ftj 

1 

CO 

i 

CO 

1 

CO 

CO 

1 

CO 

CO 

o 

o 

o 

o 

o 


o 

3 

in> 

o 




CM 

CM 

CM 







i -3 

X 

33 

0 ) 

'S 



&H 

PQ 

00 

1 

Q 

oo 

i 






*H* 

c 

o 

>. 

T 3 

>> 

T 3 

s 


* 

CO 

<D 

rH 

os 

rH 

os 

rH 

rH 

< 

rH 

rH 

*f— 


co 

E 

cd 

C 

08 

X 

fa 

cd 

as 

[3 

*C 

1 

F 

Ph 

PQ 

Q 

CM 

Q 

US 

o 

a 

• H 

1 

— !< 

(Q 

d 

a 

3 

03 


oo 

i 

os 

oo 

i 

os 

rH 

OS 

■>* 

l 

OS 

l 

OS 

o 

m 

CM 

w 

E -5 

Wil 

1 

cn 

O 

hS 

CO 

T 5 

C 

M 


*Barnes Engineering Co. 
t Block Associates, Inc. 
t Williamson Development Co. 
§Thermistor bolometer 



SPECTRORADIOMETERS 763 

By choosing arbitrarily a normalizing constant or scale factor R„, which satisfies 
R(\) = R„r(\ ), Eq. (19-2) can be written 

V~R"j P*rdk=R„P„ v (19-5) 

where 

P n = VI R n w (19-6) 

is called the normalized radiant power in the incident beam. 

19.1.3.1. Normalization to the Peak. If the choice of normalization constant is 

R n~ R(k m ax) VW _1 (19-7) 

where X mrtJ . is the wavelength at which R{\) is a maximum, as with most narrowband 
systems, then 

P" = f w (19-8) 

is called the peak-normalized power in the beam or the power normalized to the peak 
responsivity. From Eq. (19-6) and (19-7), it is apparent that this means that P„ watts 
of radiant power, if concentrated at the wavelength X max , would produce the same 
output V as the actual beam. Hence, peak-normalized power is sometimes called the 
"effective” [3] or "equivalent” power at k tna x- 
Furthermore, if the peak-normalized spectral bandwidth of the instrument is arbi¬ 
trarily defined as 


AX„ = [l/i?(X maJ -)] J R(\)d\ fjL 

(19-9) 

it may be used to compute 


P\n = P n /AX n w 

(19-10) 


the peak-normalized spectral radiant power assigned to Nnax- 

19.1.3.2. Other Methods of Normalization. There are situations where normal¬ 
ization to the peak is inappropriate because there is no wavelength A. maJ for which 
there is a single predominating'maximum value of R{\). This is more often so with 
broadband systems. It may then be more appropriate to normalize to the average, by 
choosing 

vw- 1 (19-11) 


(19-12) 

is equal to the entire wavelength interval of interest (X 2 — Xi) over which the average 
is computed in Eq. (2-11). The foregoing, and still other normalization methods and 
their implications, are discussed in greater detail in [1] and [2]. Because of the 
variety of possible normalization methods, it is important to specify clearly the method 
used when normalized values for measurement results are reported. 

19.2. Spectroradiometers 

Spectroradiometers are used to obtain an absolute measurement of the spectral 
variation of a source radiometric quantity within a very narrow waveband. Dispersing 


R 


•=[/:: 


R(K) d\ 


(X 2 — X]) 


Then the normalized bandwidth 




AX„ = (1/R„) I R(\)d\ 

A. 




764 


INFRARED MEASURING INSTRUMENTS 


elements such as prisms, diffraction gratings, or other optical elements can be used to 
produce the spectra. 

Radiometers which rapidly sequence through a set of narrow filters also can be called 
spectroradiometers. The filter is often sequenced by the rotation of a filter wheel. 
If the speed of rotation is sufficiently high, the dwell time for any one filter can approach 
the response time of the unfiltered radiometer. In such cases, the response time should 
be determined under dynamic rather than static conditions. 

The basic design of a simple prism or grating spectroradiometer is shown in Fig. 
19-4, where 

L = collecting optics 

Si = entrance slit of monochromator acting also as the field stop 
C = collimating optics 
A = dispersing element (prism or grating) 

F = refocusing optics 

S> = exit slit of monochromator 

D = detector element 



The essential components of any prism or grating spectroradiometer are the same as 
those of any radiometer (Sec. 19.1) with the addition of the following: 

(a) An entrance slit, which usually acts as the field stop of the collecting optics. 

( b) A collimator, which may be a lens or a mirror, with the entrance slit at its focus. 

(c) A dispersing element, either a prism or grating. 

( d ) A focusing element, which produces an image of the entrance slit from the parallel 
beam at each wavelength so that these images are dispersed linearly to form the familiar 
spectrum. 

(e) One or more exit slits to select the radiation in any desired region of the spectrum 
and allow it to pass on to the detector. 

A spectroradiometer can be said to be a radiometer with a monochromator located 
between the collecting optics and the detector. 

19.3. Monochromators [4] 

In self-collimating spectrometers and monochromators, a single optical element 
serves both for collimating the beam from the entrance slit and for refocusing the dis¬ 
persed beam onto the plane of the exit slit. The most frequent configurations of self- 
collimating instruments is the Littrow type shown in Fig. 19-5. In this instrument, 
a plane mirror is positioned so as to return the dispersed beam back to the prism or 






















MONOCHROMATORS 


765 


Off-Axis 

Parabolic 

Mirror 


Prism 


Plane 

Mirror 



Entrance Slit - 


Focal Plane of 
Off-Axis Mirror 

Exit Slit 



Detector 


Incident IR Radiation 


Fig. 19-5. Single-pass monochromator. 


grating for a second dispersion before it returns to the self-collimating mirror, or lens, 
and the exit slit. 

This type of monochromator has a limited resolution, determined by the finite size 
and quantity of the prism employed. An impure spectrum is also produced, particularly 
at the wavelengths greater than 8/x, because of the high-intensity short-wavelength 
light from the source being scattered into the exit slit. Various devices can be employed 
to improve purity of spectrum, such as the use of selective choppers or shutters which 
are transparent to the short-wavelength radiation, and thus only chop the desired 
longer wavelengths. The best solution can be achieved by using a double monochro¬ 
mator. If it is a Littrow type, such as the Perkin-Elmer Model 99 monochromator 
shown in Fig. 19-6, the beam is acted upon four times by the prism or grating. Not 
only is dispersion increased by these repeated dispersions but spectral resolution 
and purity are also improved. In a double monochromator, two monochromators are 
placed in series, with the exit slit of one forming the entrance slit of the second. In 
this configuration, the repeated dispersion also improves the spectral resolution and 
purity. The emerging beam from the exit slit of a well-designed double monochromator 
contains a minimum of scattered radiation of wavelengths outside the desired passband. 



Fig. 19-6. Perkin-Elmer Model 99 double-pass monochromator. 


In double-pass instruments, the second pass is made over portions of the same path 
as the first. In one type, the second-pass radiation is distinguished from the first, 
also emerging from the exit slit, only by the insertion of a chopper in a portion of the 
second-pass beam that does not overlap the first. In this case, although the modulated 





















766 


INFRARED MEASURING INSTRUMENTS 


beam from the second pass may have high spectral purity, a fairly high level of un¬ 
chopped first-pass radiation of unwanted wavelengths can also be incident on the 
detector. The effect of this first-pass radiation on the response of the detector element 
to the chopped second-pass radiation of the desired wavelengths should be carefully 
tested in such cases. This is particularly important for measurements in wavelength 
regions in which the spectral responsivity of the detector element can be very low 
as compared to its responsivity to the unchopped wavelengths from the first pass. 
Spectral filters inserted at the entrance slit can be used to produce substantial changes 
in the level of this unwanted, unmodulated radiation in order to observe the effect, if 
any, on the output. Ideally, there should be no effect, because the output should be a 
measure only of the chopped radiation in the desired wavelength band. However, this 
needs to be verified for each detector element used over the complete range of wave¬ 
lengths for which it will be employed. 

The wavelength band in the output of a prism monochromator is a direct function 
of the relative positions of the slits, prism, and any optical elements (such as a Littrow 
mirror) used to shift the dispersed spectrum across the exit slit. A grating monochro¬ 
mator, however, can have overlapping orders. A fairly low-dispersion foreprism 
monochromator, or suitable filters, can be used in front of the entrance slit to remove 
wavelengths of the undesired orders and to eliminate the ambiguities. 

19.4. Spectrometers and Spectrophotometers 

An infrared spectrometer permits selection and isolation of a desired wavelength 
(or band of wavelengths) in the infrared spectrum for study of the physical properties 
of a material. An infrared spectrophotometer permits selection and isolation of a 
desired wavelength (or band of wavelengths) in the infrared spectrum, for simultaneous 
comparative examination of the physical properties of a sample material and a reference 
material at a selected point in the spectrum. 

In spectrometer^, single-beam photometry is used for radiation measurement. The 
radiation spectrum from the source alone is first measured; then a sample is introduced 
in the sample area. The source radiation, is modified by the sample, and is then 
measured. A comparison of these two measurements, as a function of radiation fre¬ 
quency, is then calculated and replotted. In spectrophotometers, double-beam optical 
systems are used in which radiant intensity through a sample cell and a reference 
cell is compared and automatically recorded. In spectrometers and spectrophotometers, 
the spectrum is usually dispersed by a monochromator employing a refraction prism or 
a diffraction grating. 

19.4.1. Infrared Grating Spectrometry [6]. Infrared grating spectrometers are 
used for studies of molecular structure, where rapid automatic recording and extremely 
high resolution are required. When used with multiple reflection cells, long path-length 
studies in gases and liquids can be made. Fast recording is achieved by using highly 
sensitive lead sulfide, lead telluride, lead selenide, or indium antimonide detector 
cells with rapid response times. Improved gratings and recent improvements in detec¬ 
tors and in amplifier design have made possible the construction of instruments with 
resolving powers exceeding 150,000. 

Optical systems are generally of the off-axis parabolic mirror type, or the on-axis 
Pfund type. Mirrors of high quality are used to obtain good images and high resolving 
power. The Pfund type employs on-axis paraboloid mirrors and plane mirrors with an 
aperture in the center to produce superior images and higher resolving power. 

19.4.1.1. Off-Axis Parabolic Grating Spectrometer [7]. An off-axis type, double¬ 
pass instrument is shown in Fig. 19-7. It has a focal length of 10 m, achieved by 
mounting the grating in the fashion of the Littrow instrument to ensure double passage 


SPECTROMETERS AND SPECTROPHOTOMETERS 767 

of the radiation beam through the optical system. Using a grating of great perfection, 
a resolving power of 120,000 to 140,000 lines per inch in the 1.3- to 1.7-/U, region is 
obtained, with an increase in resolving power of about one order of magnitude at longer 
wavelengths. 




19.4.1.2. On-Axis Pfund Grating Spectrometer [6]. A typical on-axis Pfund-type 
direct-recording spectrometer is shown in Fig. 19-8. Incident infrared radiation focused 
by a collimating lens on the entrance slit and modulated by a chopper passes through 
the central aperture of plane mirror M,. Reflected by the paraboloidal mirror, Pi, it 
emerges as a parallel beam of radiation, which is reflected by mirror M x to the grating. 
The grating is accurately located on a turntable, which may be rotated in order to scan 
the spectrum. From the grating, the diffracted beam, reflected by mirror M 2 , is focused 
by a second paraboloid, P 2 , through the central aperture of mirror M 2 to the exit slit. 
The emerging beam is then focused by the ellipsoidal mirror, M. t , on the detector. This 
type of on-axis system produces a better spectral image and superior resolution. 

19.4.2. Commercially Available Spectrometers and Spectrophotometers. Table 
19-3 lists some of the characteristics of some commercially available spectrometers and 
spectrophotometers. Representative instrument types are described in the following 
paragraphs. 



















Table 19-3. Characteristics of Commercially Available Spectrometers and Spectrophotometers 


768 INFRARED MEASURING INSTRUMENTS 


>50-0 

c o 
4! 3 
S; "o 
o £ 
^ & 
QS 


3 

O 

o 


C 

„ ^ co 

S CD 

o 


Q 

£ §5 


c 

o 


o 

co 

a: 


c 

•2 

I 

O' 


| 

s 

o 

o 

5 

c 

o 


o 

c 


cu 


CO 

oi 

w 

H 

W 

S 

o 

« 

H 

O 

w 

&- 

CO 


*2 
CM ^ 
H> CO 
CO w 

5*3 

^co 

O 

o o 
o o 


c 

6 

CM 

CM 


lO 

I 

CM 


=t 

CM 

*-H 

CO 

4 

rH 

o 

o 

V 


£ 

CO 

0) 

X 


be 

C 

m 


iO 


£ 

o 

H 


CO - 

co C 

i- 

a 
at _ 

3 o 

3 at 
O Z 
Q 


m 

oo 


o 

CM 


»- 
- o 


6 


co A 
to y 
at e 

OP 

^ *h 

o 

3 h 
o .c 

^ £ ^ 13 

C co 0) 

CO bo CO CO 

y c bo^ 


CO 
o 
o ^3 

(m -r- 

•H co 


CO X 
• -*-> 

O. 3 
« C 

03 


6 

tn 

'C 

a 


c _ 

S .-o 

o b « 

+3 t. Z 


Qu 

i- 

o 

O 

t* 

at 

B 

S 

i 

c 

2 

u 

<D 

Oh 


CM 


c 

co 

CO 


a 

cO 

Dc5 


co 

i 

IO 

CO 


X 

CO 

£ 

I 

£ 

o 

o 


CD 


X 

CO 

£ 

i 

£ 

CJ 

O 


o 

C 


be 

C 

‘C 

a> 

a; 

c 

*bb 

a 

w 


CQ 


E- 


CM 

i 

co 


2L 

rH 

o 


o _ 
U at 
• c 
50 c 
C at 


W 

CO 

O) 

C 

H 

cO 

OQ 


3 iO 


s 

w 

H 

w 

S 

o 

o 

X 

CL 

o 

« 

H 

Q 

w 

CL 

co 


O 

iO 

i 

o 

a> 


4 4 

E S a 

o o 0 i 
w 00 R 0 

CM IO CM 2 

H> H> H> 

co co co q 
4 4 4 ^ 
EEEn 
cm \o o o 
d rH ic d 


« a c 

e =3 10 I 

I g 3 -8 >. 

- S Tl ® -S 

a § m S 

.2 '-33 -S 

CO CO 

2.S g 

- « E s 

u g -g 1 ; 

c» 


.8 

3 

o 

T3 


0) 

c 


L- 

^3 


IO 


^ a 

£3 

J 2 

co be 
co a 

at c 
g> o 

•£ 6 
co 


o -t 

+1 +1 


4 

co 

<=> ^ 
o ^ 
+1 +1 


w 

CM 

I 

lO 

CM 


CO 

_o 

■X 

a 

o 

£ 

CO 

0) 

X) 

JD 

2 =3 

S c 


uo 


(h 

CQ 

£ 

CO 

’C 

a 

o 

O 

t> 


o iO 

o o 

+1 +1 


O rH 

d o 
+1 +1 


io 


iO 

CM 


CO 

o 


a 

o 

£ 

CO 

O) 

X 

J2 

2 =3 

S 3 

a c 


io 


O 

CO 

2 

£ 

CO 

c 

a 

o 

O 

I> 


a 

u 

o 

O 

Vh 

cy 

£ 

3 

b 00 

Oh 


t> 

CO 

rH 

Sh 

03 


CQ 

I 

CO 




























Table 19-3. Characteristics of Commercially Available Spectrometers and Spectrophotometers ( Continued ) 


SPECTROMETERS AND SPECTROPHOTOMETERS 769 


* 


< 50-0 

I! 

a> -« 

o Cl 

a ” 
$ 

os 


a. 

m 

S £ 

o 

© d 
+1 +1 


*£> 

o ^ 

O lO 

o d 
+1 +1 


cu 

sl 

c £ 

a. a. 

oo m 
o o 


O O 3 
MO" 1 
hh t- <N 
CO 

a. 

•<r 


CO cfl 

a. a. 


o 

o 

1.5 

3.0 

3, 

m 

a. 

a. 


o 

o 


-*-> 


CM 

G 

-*-> 

-*-> 

a. 

G 

G 

CO 

a. 

a. 

o 

o 

CM 

d 

CM 

CM 


a. 

W 

o 

o 

d 


o 

d 


I 

3 

8 


a. 

<N 


c> o 
+1 +1 +1 +1 


a> 

>3 

Id co 
c > 

i.S 
m a. 

d —i 

+1 +1 


a. 

in 

i-H 

o 

o 


a. 

in 

CM 

O 

o 


c 

c “o S 

m G co 
y oi 
O r\ 


C 

E 

CM 

u 

O 

00 


c 

E 

00 

u 

o 

<M 


C 

£ 

<M 

"G 

G 

CO 

00 


G 

a 

o 


o 

g 

CO 

in 


£ 

o 

(M 

CM , 

o < 

P j 


~Q 

2 

•fi 

CO 

> 


G C 

■ • »H 
£ £ 
co m 


a m 

8 §5 


m 

CD 


oo 

00 


o o 

05 m 
00 t> 

o o 

i i 

o o 
05 m 

*-H CO 

o d 


i 

in 
cm in 


CD 


£ 

CJ 

o 

co 

I 

o 

CD 

CD 


CM 

i 

m 

r- 


CM 

x 


m 

CM 


no 

G 

3 

J 2 

3. 

■ 

CM 

CM 


m 


CD 

i-H 

I 

CM 


£ 

o 


o 

co 

G 

ft: 


=i 

T3 

G 

O 

>» 

£ 

7 7 

£ £ 1 E 

o o cq o 
T}< o ^ o 


a. a. a. 
£ £ £ 
o o o 

<-* 05 O 
<M 00 CO 
<-> 

A CO A 

a. a. a. 
£ £ £ 
<n -cj< m 
o o -h 


E 

o 

t> 

o 


a. 

£ 

in 3. 

<M 
cO -*-> 

a. « 
c a. 
« 6 
O in 

o -i 


I 

£ 

u 

CO 

o T 
c £ 

Cfl S 

?g 

5 S3 

I « 

CQ 


< d 
O 3 

00 S 

c S 

cfl J 

JS r- 

00 d 

a * 


a. 

o 

rH 

CO 

a. 

O 

o 


05 

-*-> 

CO 

£ 

O 

f-H 

CM 


£ 
O 
■ **» 

1 

o 


CO 

o 

a 

o 

£ 

CO 

0 ) 

~Q 

G 

|3 
p c 


cO 

o 

-4-> 

a 

o 

E 

3 

JD 

3 

g G 
O c 

Q C 


g 

o 

-G 

a 

o 

£ 

3 

£ 

^g 

2 

3 G 
O c 

Q 


E 

CO 

0 ) 

JO 

3 

G 

o 

a 


E 

cO 

3 

jo 

3 

3 

G 

O 

Q 


G 

o 

-G 

a 

o 

£ 

G 

0 ) 

-O 

JL5 

2 =3 

g G 

Q C 


G 

CJ 

G 

a 

o 

£~ 

G 

g 

G 


u 

G 

Q) 

C 


a 

o 


o 

0 ) 


E 

G 

G 


u 

JO 2 
^ cG 

a 

§ 3 C S 
Q c cn 03 


S 

9 E 

m 9 

£ ij = 

jj .. ® g 

o —. >- 

3 -r M — 
§ £ C CO 
Jr C o 

Q co 


o 

c 


m 


m 


m 


°o 

co 


o> 


CM 


2 

a 

S 

8 

■< 


e 

o 


I 


be 

a 

d 

CO 

be 


-o 

3 

O 

P 


o £ 
co fe 

'C 

S a 
fc « 

• 3 o 

e s 

? 00 

£ "2 
D cfl 

d a 

j 


be 

C 

"co 

be 

_oj 

3 

3 

O 

Q 


o 

CM 

3 

’S 

s 


A "3 

c SI 

O 3 

E 52 

> o 

S « Cfl 

d c £ 
; o .2 

G Cl 

jh 6 cs 

-Q 2 -• 
3 E 
^ O 


o 

'to 


cfl 

bo 

C 

% d 

£ 2 
d be 
dM 
J3 

S '? 

05 G 

'bo § 

.s i 

05 


d cfl 
« Z 
£ Z 
O C! 

5 £ 

1 s 
c 
o 

£ 

® x 

-O 
3 

O 

Q 


m 

i> 


o 

co 


£ 

00 

C 

a 


i_ 

3 

G 

£_ 

£ o 

-C « 

8 Z 

c c 

2 E 
£ .2 
ja S. 

boo 

d co 
co 


S 

u 

H 

W 

S 

£ 

O 

X 

CL 

o 

£ 

H 

O 

u 

£ 

CO 


o 

CO 

"O 

G 

Im 

G 


V 

G 


G 

G 

E 

G 

CO 

G 


o 

I 

co 


CM 


Sh 

O 

O 

CO 

G 

m 

CM 

CM 

Ul 

co 


CM 


G 

d 

o 

Oh 


C 

Cfl 

s ^ 
2 s 

m 


o 

rt 

Z 

< 

in 


























Table 19-3. Characteristics of Commercially Available Spectrometers and Spectrophotometers ( Continued) 


770 


INFRARED MEASURING INSTRUMENTS 


fl 

S' 13 

^ §■ 
03 


o 

d 


i 

£ 

u 

to 

<N 

o 


__ 03 

TJ b 
C 43 
as "p 

IS o 

C ^ 
.il eg 


c 

j: 

-w 

Tf 



*T3 


1 

d 

ao 

E 

CO 

h 

0) 

O 

V 


o 

i- 

co 

fcd 

o 


£ 

u 


«o 

o ^ 
o —< 

o d 
+1 +1 


£ 

g 

3 

O 

o 


O 


u £ b b 

- 5 - -£ - 43 03 

7 - V - t3 -a 

I ^ I M I - t. 

Eo£°£o7 o 

O -O y -o C x 

» 5 P <N « « ° 3 

O <N <M lO 


s 

o 

iq 

o 


O lO 

d o 
+1 +1 


co U 

- E 

I W „ 

£ o 
o o io 
't g o 
+1 w Xl 


+1 


e 

c "o 8 

~ CO 

y 

O 


c c 

•H • H 

E £ 

00 w 


b 

J3 

o 

o 

o 

r-H 

o 

-b 

c 

£ 


A 3. 

.£ j~ 
E- 


c 

£ 

o 

<M 


a 

d »- 
8 a . 


iO 

CO 


lO 


iO 

csi 


6 

cj 

o 

ic 

rf 

6 

o 

o 


ao 

c? 

1C 

(M 


<N 

CN 

<N 


C 

o 


o 

co 

<*> 

ft; 


S 

CJ 

C • 

^ I 

5 « 

« tfl 
CQ 


* 7 

~ E 

£ « 
o o 

§2 

o 


s- 

o 


i $ 

lO +* 

a> 

• JO 

o 


E 

o 

£ i 

25 


co 

\ I 

o u 

« § 

o 


c 

•2 

■*«» 

§ 

O 


CO 

CJ 


a 

o 

6 

CO 

0) 

JQ 

a> 


■w 
8 
% 

S 

a 

J8 =! 

--J8 c 


!- 

CO 

<D 

c 


-§ ~ be _ 
o 2 c co 

JSr C • >-* o 

a co 


CO 

o 
*Z3 
a 
o 

S 

3 
$ 
x> 

0) 

Esr-o. 

2 1 “I 

C ^ CO 


E 

« . 
J « 

•“ <C 


o 

Q 


E 

CO 

0) 

JO) 

3 

o 

Q 


cd 

o 

*■♦3 

a 

o 

S 

CO 

a> 

jo 

<D 


CO 

c 

o 

-*-» 

a 

o 

S 

CO 

0) 

-O 

0) 


CO 

>3 

a 

o 

B 

CO 

<D 

a) 


3^ M 

O 2 C 
D C co 


CO 

c 

o 

a 

o 

E 

cfl 

a> 

-o 

0) 


•§s M 
O 2 C 
D c co 


o 

c 


U5 


k 

S 

a 

fc 

g 

-e 

8 

e 

I 


<y 

£ 


u* 

o 

CO 

E 

8 o 

X CO 

8 Z 

I 6 " 

C w 
O Q, 
bo o 
d ^ 
•S ^ 
CO 


rtl o 
C£) 

•§ ® 

^ * 

5P * — 

.S EO 

g|l 

?1 £ 
E c s 
® o •- 
n C £ a 

IX 


-o 

3 

O 

TJ 


O 

u 


-I 

J 

fc) J3 

E c 
o 

E 


CO 


b 

a. 


E 

ao 

C 

a 

u. 

CQ ■ 


E 

03 

’fc 

a 

o 

O 

CO 

B“ 

£ 

o 

oo 

X 

o 

o 


b 

<13 

a 

b 

3 

H 

>> 

C (30 
0) 

S3 ^ 
O CO 

■8 “ 
cn £ 

33 ® 

2 & 


2 

Cd 

H 

M 

s 

8 

o 

I 

a. 

O 

as 

H 

O 

u 

CL 

03 


CQ 

03 

u 

< 

lO 

I 

QS 


OS 


.c o 

^ 2 
i. 43 

& 'S 

C§ S 


o 

C 


E 

3 

< 

-a 

b 

'3 

CQ 




00 

Z 

















SPECTROMETERS AND SPECTROPHOTOMETERS 


771 


19.4.2.1. Single-Beam Double-Pass Spectrometer. An example of a single-beam 
double-pass infrared spectrometer is the Perkin-Elmer Model 112 instrument shown 
in Fig. 19-9. Infrared radiation from a source is focused by mirrors M\ and M 2 on the 
entrance slit, Si, of the monochromator. The radiation beam from Si, path 1, is col¬ 
limated by the off-axis paraboloid, M ;J , and a parallel beam traverses the prism for a 
first refraction. The beam is reflected by the Littrow mirror, M 4 , through the prism 
for a second refraction, and focused by the paraboloid, path 2, at the corner mirror, 
M«. The radiation returns along path 3, traverses the prism again, and is returned 
back along path 4 for reflection by mirror M 7 to the exit slit, S 2 . By this double dis¬ 
persion the radiation is spread out along the plane of S 2 . The frequency interval which 
passes through S 2 is focused by mirrors M H and M» on the thermocouple, TC. The beam 
is chopped by CH, near A/«, to produce an ac voltage (at the thermocouple) which is 
proportional to the radiant power or intensity of the beam. This voltage is amplified 
and recorded by an electronic potentiometer. Motor-driven rotation of Littrow mirror 
M a causes the infrared spectrum to pass across exit slit S 2 , permitting measurement of 
the radiant intensity of successive frequencies. 



19.4.2.2. Prism-Grating Double Monochromator Spectrometer [6]. This instrument, 
manufactured by Unicam Instruments, Ltd., uses a prism-grating double monochro¬ 
mator covering the widest possible spectral range without breaks for optical changes 
or adjustments. The instrument is shown in Fig. 19-10. 

The prism monochromator has four interchangeable prisms, and the grating mono¬ 
chromator has two interchangeable gratings. The two monochromators, gaged by cams 
linear in wave number, are driven by a common shaft. The instrument may be used 
either as a prism-grating double monochromator, or as a prism spectrometer by blank¬ 
ing the grating monochromator. Gratings, prisms, and cams may be automatically 
interchanged by means of pushbuttons. Magnetically operated slits, programmed by 
a tapped potentiometer, provide a constant energy background. A star wheel time¬ 
sharing beam attenuator is used in the double-beam photometer. 

































































772 


INFRARED MEASURING INSTRUMENTS 


Pfund Condenser 



Fig. 19-10. Unicam prism-grating double monochromator spectrometer. 



Fig. 19-11. Infrared flame-temperature 
spectrometer. 


19.4.2.3. Flame-Temperature Spectrometer [6]. This spectrometer provides a meth¬ 
od to measure the temperatures of missile flames up to 5 ft in diameter up to several 
thousand degrees centigrade. The instrument, shown in Fig. 19-11, is rapid and 
accurate, and requires no calibration or attenuation of the gas stream. 

Infrared radiant energy from a source of known emission passes through the hot gas 
stream with absorption bands due to the water vapor and carbon dioxide present in the 
gas stream as products of combustion. The infrared beam is focused on the entrance 
slit of a Perkin-Elmer Model 98 monochromator. At a given wavelength, with the 
shutter out of the beam, the radiant energy E x is measured: 


Ei = E s t + E ( {\ — t) 


(19-13) 
























































SPECTROMETERS AND SPECTROPHOTOMETERS 


773 


where E s = known energy from the source 

E, = energy from blackbody source at gas temperature T 

t = unknown percent transmission of carbon dioxide and water vapor in gas 
stream 

With the shutter in the beam, the energy E> is measured: 

E 2 = E, (1 — t) (19-14) 

The two equations are then solved for r and E, and the apparent temperature of the gas 
stream is obtained. 

19.4.2.4. Rapid-Scan Spectrometer. The rapid-scan spectrometer manufactured 
by Perkin-Elmer Corp. (Fig. 19-12) records the distribution and time variation of the 
spectral wavelengths of the energy radiated during the powered flight portions of missile 
firings. The instrument consists of a rapid-scan monochromator, a radiation detection 
system, and appropriate readout and recording equipment. 


Lead Sulfide 

Multiplier 
Phototube 


Off-Axis 

Paraboloid 

Mirror- 



8" Aperture 
40" Effective 
Focal Length 
Collecting Optics 


Pass 
Upper Level 


Diagonal 
Mirror 

Roof Mirror 


Nutating Mirror 


Fig. 19-12. Perkin-Elmer rapid-scan spectrometer. 


The rapid-scan monochromator utilizes a double-pass system where the first and 
second passes are physically separated. As shown in Fig. 19-12, radiant energy from 
the source is imaged by the collecting optics, composed of a modified double-pass Littrow 
system, on the entrance slit of the monochromator. The beam is collimated by an 
off-axis paraboloid mirror onto the prism. The energy beam is then refracted to the 
nutating mirror, reflected to the roof mirror (where the vertical light motion com¬ 
ponent is eliminated), and then back through the prism system. The returned beam 
is brought to a focus by the paraboloid mirror at the corner cube mirror, where it is 
displaced up and across, back to the paraboloid mirror. It then passes through the 
monochromator for the second time. The second pass is 1 in. higher than the first pass, 
and the second pass only is intercepted by the diagonal mirror and focused on the 
exit slit. 

The detector sees an alternating signal whose frequency is the chopped frequency; 
within each half cycle the signal amplitude varies in accordance with the radiant 
spectral energy. The result is a plot of the spectrum of the source repeated at the 
chopping frequency. 

















774 


INFRARED MEASURING INSTRUMENTS 



Fig. 19-13. Barnes in-line spectrometer. 


19.4.2.5. In-Line Spectrometer. The lens configuration in the Barnes instrument 
(Fig. 19-13) permits effective use of noncollimated light with reflection gratings. The 
major advantage of using converging light in spectrometers is that it permits con¬ 
struction of an in-line instrument. In this technique, light rays received at one end 
travel straight through the spectrometer. The rays are then filtered into narrow wave 
bands and are dispersed into a useful spectrum. They may then be photographically 
or electronically recorded at the opposite end. 

The in-line spectrometer uses a correcting prism placed ahead of the refraction 
grating. The prism is designed so that its differential refracting effect on rays of 
various inclination exactly compensates for the difference in light-path length. Thus, 
all rays contributing to the resulting spectrum line will focus in one plane. 

19.4.2.6. Interferometer Spectrometer. The Block Engineering interferometer-spec¬ 
trometer has greater throughput than conventional spectrometer, because the inter¬ 
ferometer has a large entrance aperture determined by the mirror size. This enables 
the instrument to accept more radiant energy from the source than prism or grating 
instruments in which the entrance aperture is limited by narrow slits. 

High sensitivity gain is due to the instrument’s examining each wavelength through¬ 
out the entire time period of each scan. In a conventional instrument, each wavelength 
is examined for only a very short part of the scan time (1/rcth the scan time if n is the 
number of resolution elements); the interferometer achieves a gain which is Vn for 
the same scan time. For typical instruments this can be a factor of 50. Furthermore, 
this gain is realized even when one examines point sources where the throughput gain 
is not large. 

A block diagram of an interferometer spectrometer is shown in Fig. 19-14. Incoming 
infrared radiation is received by the interferometer, and a fringe pattern is produced. 
When one of the mirrors in the interferometer is moved back and forth at a slow, con¬ 
stant velocity, the motion is manifested as an alternate brightening and darkening of 
the central fringe. 



Fig. 19-14. Block Associates interferometer spectrometer. 




























SPECTROMETERS AND SPECTROPHOTOMETERS 


775 


An infrared detector placed at the central fringe converts these cyclic changes into 
an ac signal. If the mirror velocity is kept constant at a predetermined value, the 
frequency of the ac signal from the detector is directly related to the wavelength of 
incident radiation, assuming that the incident radiation is at one given wavelength 
(monochromatic). 

If another wavelength twice as long as the first (half the frequency) should be sub¬ 
stituted as the incident radiation source, the ac output signal from the detector would 
be at one-half the frequency of the first. The amplitudes of the two signals would 
remain the same if the maximum brightness of the two sources were the same. 

If incident radiation containing many wavelengths were introduced into the system, 
the output of the detector would consist of a superposition of all the ac signals which 
correspond to all the wavelengths in the source. 

The output of the interferometer system is tape recorded and played back through an 
audio wave analyzer to recover the infrared spectrum. 

The scan drive must be linear and constant and the effects of source-intensity varia¬ 
tions must be negligible. Either of these can cause apparent spectral peaks in the 
output. 

19.4.2.7. Double-Beam Optical Null Spectrophotometer. An example of a double¬ 
beam optical null system is the Beckman spectrophotometer shown in Fig. 19-15. 
The instrument utilizes a double monochromator which is convertible to single-beam 
operation. 



Fig. 19-15. Beckman IR-4 automatic recording infrared 
spectrophotometer. 

For double-beam operation, radiation from a Nernst glower, N, passes through the 
half mirror, C\, rotating at 11 cps. The radiation then passes through the sample 
during one half revolution of the mirror, where it is recombined by the symchronously 
rotating half mirror, C 2 . During the other half revolution of the mirror, the beam is 
deflected through the reference. Mirror C\ chops the radiation beam at 11 cps and 
directs it alternately through the sample and through the reference. 

The recombined beam is then directed through the controllable entrance slit, Si, 
into the first of two monochromators. In the first monochromator, the beam is transmit¬ 
ted by a collimator, Cob, through a prism, Pi, and associated Littrow mirrors for initial 
dispersion. This dispersed beam then passes through fixed-width entrance slit S 2 , 
into the second monochromator. Here collimator Col 2 transmits it through prism P 2 
and Littrow mirrors for further dispersion. 

The doubly dispersed monochromatic beam now exits through the controllable exit 
slit, S 3 , and is focused on a thermocouple, T. The signal from the thermocouple is 
amplified and used to position an optical attenuator, A, in the beam path, so that 
the radiation transmitted by sample and reference beams are equal in intensity. The 
position of the optical attenuator determines the position of the recorder pen, producing 
linear wavelength records. 













776 


INFRARED MEASURING INSTRUMENTS 


For single-beam operation, the reference beam is blocked. The thermocouple re¬ 
ceives only energy from the sample beam, chopped at 11 cps. The amplified thermo¬ 
couple output is recorded directly by the pen. 

19.4.2.8. Direct-Ratio Spectrophotometer. A direct-ratio system is used in the 
Perkin-Elmer instrument (Fig. 19-16). This is the same as the optical null system in 
that the two radiation beams are simultaneously compared. In this case, however, 
a separate electrical signal proportional to each beam is separated by a phase-sensitive 
detection signal. The reference beam is placed across the slide wire of a recording 
potentiometer and the sample signal fed to a standard strip chart recorder so that the 
ratio between reference and sample beams is automatically recorded. The optical 
system of the instrument is shown in the figure. 



Fig. 19-16. Perkin-Elmer universal spectrophotometer Model 13-U. 


19.5. Interferometers 

Interferometers divide a beam of light into two or more parts which travel different 
paths and recombine to form interference rings. The form of these rings is determined 
by the difference between the optical paths of the successive beams. Interferometers 
measure the difference in optical path length and refractive index. 

19.5.1. Rayleigh Interferometer [7). In the Rayleigh interferometer shown in Fig. 
19-17, monochromatic light from linear source S\ falls on a screen and is split into 
two beams by well-separated slits, S 2 and S 3 . The light is rendered parallel by lens 
L, and passes through two exactly similar tubes, closed at each end by transparent 
windows. The contents of the tubes (liquid or gaseous) determine the positions of the 
fringes by the path difference introduced between the two beams. The beams leaving 
the tubes are focused by lens L 2 , producing an image whose bandwidth is determined 
by the width of S 2 and S : , and by the magnification of the system. 



Fig. 19-17. Rayleigh interferometer. 








































INTERFEROMETERS 


777 


Mirror M^ 



Eye 


Fig. 19-18. Michelson interferometer. 


19.5.2. Michelson Interferometer. The Michelson interferometer, shown in Fig. 
19-18, consists of two plane mirrors, Mi and M 2 , one of which is adjustable, and two 
plane-parallel plates, G\ and G 2 . Light from an extended source is incident at 45° on 
plate G i, (partially silvered on the rear surface) and is divided into reflected (path A) 
and transmitted beams (path B) of equal intensity. The light reflected from mirror M\ 
passes through plate Gi a third time before it reaches the eye. The light reflected from 
mirror M 2 passes back through G 2 a second time, is reflected from the surface of plate 
Gi, and into the eye. The two beams have a phase difference governed by the difference 
in the two paths. 

Compensating plate G 2 compensates for the passage of light through the plate in 
path A. Its use is not essential for producing fringes in monochromatic light but is 
indispensable when white light is used. The light from every point on the extended 
source interferes with itself according to the distance between mirrors, or according 
to the different length of arms A and B. Constructive interference will occur when 

2dcos0=raX (19-15) 

where d = path difference 

6 = angle to a source element imaged by Mi or M 2 as seen by the eye 

19.5.3. Twyman-Green Interferometer 171. In 
the Twyman-Green interferometer, a Michelson 
interferometer is illuminated with strictly parallel 
monochromatic light, produced by a point source at 
the principal focus of a well-corrected lens. As 
shown in Fig. 19-19, light comes from a pinhole, P, 
at the principal focus of a lens, L\. A second lens, 

L 2 , focuses the emerging light into the eye. By 
the use of collimated light, the fringes at infinity 
can be seen at a finite distance (such as on one of 
the surfaces of a prism) because of the greatly in¬ 
creased depth of focus. In addition, the light is 
made to traverse the optical part under test (a prism 
in Fig. 19-19), making the result of the test explicit. 



■¥ 


Fig. 19-19. Twyman-Green 
interferometer. 
























INFRARED MEASURING INSTRUMENTS 


778 



Fig. 19-20. Fabry-Perot interferometer. 


19.5.4. Fabry-Perot Interferometer [8]. The Fabry-Perot interferometer, shown 
in Fig. 19-20, consists of two quartz or glass plates, one face of each worked very flat 
(l/20th-l/100th fringe), partially aluminized, and separated by a spacer which renders 
the two faces exactly parallel. The outer faces form a small wedge angle with the 
working faces to prevent spurious fringes. Any entering ray will be reflected back 
and forth between the faces, and at each reflection some radiation will be transmitted. 
Some five to thirty successive rays which result from a single incident ray will be 
transmitted. These reinforce each other if the path distances between the two internal 
faces are integral multiples of the wavelength. The condition for obtaining a bright 
fringe is given by nk = 2 1 cos 6 = 2 1, approximately, since 6 is small. The pattern pro¬ 
jected on the slit of the spectrograph is a series of concentric, circular interference 
fringes governed by the relation 


2 1 . 2 1 

n =-1- 

X X 



where n — order of .interference at the center of the ring system 
t — separation of the plates 
D = diameter of a particular ring of wavelength 
f = focal length of the lens which projects the fringes on the slit 


(19-16) 


The term 2 t/k represents the integral order of interference of the innermost ring, 
and the last term of Eq. (19-16) represents the fractional order between a ring of diam¬ 
eter D and the center. 

The quantity t must first be determined by measurements on a known wavelength 
standard. Thereafter the wavelengths of unknown lines can be measured to thou¬ 
sandths of an angstrom, if the lines themselves are sufficiently sharp. 

As the thickness of the aluminized or multiple-layer coating increases, the number 
of the reflections, and hence of the interfering rays, increases, so that the fringes become 
sharper. Absorption also becomes greater, causing the intensity of the pattern to 
decrease. Reflectivity is generally considered the best compromise between high 
resolution and low intensity. Quantitatively, the resolving power is given by 


X _ 27m Vr 3 Vr 

dk V4.45 (1 — r) ° 1 — r 


(19-17) 


where 3 Vr/ (1 — r) can be considered physically as a measure of the effective number of 
interfering beams. 

The Fabry-Perot interferometer achieves the highest resolution of any known optical 
element. Wavelength measurements can be made to thousandths, or even ten thou¬ 
sandths, of an angstrom, and the instrument is used almost exclusively for ultrahigh 
resolution. 





















INTERFEROMETERS 779 

For a Fabry-Perot interferometer, the free spectral range, expressed in angstroms, 
is given by Fx = \ 2 /2 1 , so that for Hg ( = 5461 A), 

(5461) 2 

Fx =-= 0.15 A (19-18) 

2 x 10 8 

Thus, a high-dispersion high-resolution spectrograph is required to avoid overlapping 
orders in the Fabry-Perot pattern. In practice, a diameter of the ring pattern of the 
etalon is focused on the slit of a large spectrograph and some overlapping is tolerated 
for lines closer than 1 to 5 A. 

19.5.5. Spherical Fabry-Perot Interferometer [9]. At resolutions of several 
million, the luminosity of the Fabry-Perot interferometer decreases and sources at 
resolutions in this order are inherently weak. The spherical Fabry-Perot interferom¬ 
eter permits considerable progress in the study of phenomena in the area of high re¬ 
solution. 

The instrument, shown in Fig. 19-21, consists of two spherical mirrors whose separa¬ 
tion is equal to the radius of curvature r, so that the paraxial foci coincide and the 
instrument is an afocal system. One half of the surface of the mirrors is semi-reflecting 
and the other half is fully reflecting. Any incident ray gives rise to an infinite number 
of outgoing rays which are coincident, and not only parallel, as with the plane Fabry- 
Perot instrument. Neglecting aberrations, their path difference is A = 4r, a constant. 
This requires that both mirrors be stopped down with circular diaphagms of diameter 
d. It can be shown that d increases with r, and that the light-gathering power is pro¬ 
portional to r (and thus to the theoretical resolving power R 0 instead of being inversely 
proportional to Rn). For practical useful values of r (a few centimeters), d always 
remains quite small (a few millimeters), permitting relatively easy figuring of nearly 
perfect spherical plates of this size. 



Fig. 19-21. Spherical Fabry-Perot interferometer. 

19.5.6. Lummer-Gehrcke Plate. The Lummer-Gehrcke plate (Fig. 19-22) utilizes 
the interference between successive reflections in a thin quartz plate. Light is intro¬ 
duced to the plate by a total reflection prism, L,. It then undergoes multiple internal 
reflections very near the critical angle of total reflection. The beams emerging at a 
grazing angle are brought to interference by a second lens, L 2 . High reflectance and 
resolving power are thus obtained with unsilvered surfaces. 


Lummer-Gehrcke Plate 



Fig. 19-22. Lummer-Gehrcke plate interferometer. 





















780 


INFRARED MEASURING INSTRUMENTS 


19.5.7. Spectral Transmittance of Interferometers [4]. The spectral transmit¬ 
tance functions, r(X) of the Fabry-Perot, Lummer-Gehrcke plate, and Michelson inter¬ 
ferometers are described below. 

Fabry-Perot interferometer: 


T (X) 


i7 2 /4p 


1 -I- T 9 2 sin 2 77 p 


(19-19) 


where 17 2 


4p 

(1 - p) 2 


and 


p = the reflectance of the coatings 
p = the order number 


AX = 


X 2 
277 't 


1 ~P 
P 


where t = spacer thickness 
Lummer-Gehrcke plate: 


(19-20) 


(1 — p v ) 2 + 4p v sin 2 (Tr 8 N/k) 
(1 — p) 2 + 4p sin 2 ( 77 - 8 /X) 


(19-21) 


where N = number of emergent beams obtained for a given plate length and angle of 
emergence 

8 = path difference between two successive beams 


Michelson interferometer: 


t (X) ~ F\ • F 2 ‘ F 3 


(19-22) 


where F\, F 2 are factors for the Fraunhofer patterns obtained with rectangular objects 
(they are of the (sin 2 /3)//3 2 form) 

Fi describes the interference pattern due to the N steps of the echelon 
Figure 19-23 shows the spectral transmittance in a Michelson interferometer, where 

a = amplitude of initial incident radiation at B 
a/2 = amplitude of radiation in paths BM X and BM 2 
a' = amplitude of exit radiation 
Mi, M 2 = mirrors 

B = beamsplitter 



Fig. 19-23. Spectral transmittance 
in Michelson interferometer. 












REFERENCES 


781 


The amplitude, a, of an incident wave can be considered to be split equally into two 
parts by the beamsplitter, B, neglecting factors such as phase shift on reflection, the 
need for compensation plates, and other departures from the ideal case. Assuming 
Mi and M 2 to be perpendicular, the divided amplitudes a/2 are associated with the two 
paths, BM i and BM 2 ; the amplitude of the emergent wave, a', is the result of the vector 
addition of one-half of each of these amplitudes. The emergent intensity distribution 
is obtained by a '* a', where a'* is the complex conjugate of a'. Thus the transmittance 
for a Michelson interferometer for wave fronts which ara parallel to Mi and M 2 is 

t (X) ~ a 2 cos 2 (8/2) (19-23) 

where 8 = 4nt/k 

t = relative displacement of M ( from the zero phase position, i.e., from BM i = BM t 

In the absence of multiple reflections, the interference fringes obtained are quite 
broad as compared to those from a Fabry-Perot interferometer. However, high re¬ 
solution can still be obtained with long path differences if the problem of overlapping 
orders can be easily solved. If the movable mirror is oscillated in a sawtooth wave so 
that the mirror moves at a constant velocity for an appreciable portion of a wave cycle, 
then the transmission of the interferometer also varies for each wavelength. The 
output frequencies of such an interferometer-spectrometer are directly related to the 
wave numbers present in the incoming radiation. Wavelength identification is not 
difficult in this method of operation. On the other hand, Fabry-Perot interferometers 
used with large spacers require careful order sorting. Graphical means exist which can 
greatly simplify this problem. 

References 

1. M. R. Holter et al., Fundamentals of Infrared Technology, Macmillan, New York (1962). 

2. Infrared Target and Background Radiometric Measurements: Concepts, Units, and Techniques, 
Report of the Working Group on Infrared Backgrounds (WGIRB), Rept. No. 2389-64-T, Institute 
of Science and Technology, The University of Michigan, Ann Arbor, Mich. AD 289 810. 

3. H. I. Sumnicht, "Infrared 'Effective’ Radiation,” Proc. IRIS, IV, 1, 52 (1959). 

4. F. E. Nicodemus and G. J. Zissis, Methods of Radiometric Calibration, Rept. No. 4613-20-R, 
Institute of Science and Technology, The University of Michigan, Ann Arbor, Mich. (October 
1962). 

5. Techniques, Barnes Engineering Co., Stamford, Conn. (Spring 1956). 

6. H. L. Hackforth, Infrared Radiation, McGraw-Hill, New York, (1960), Chapter 9. 

7. J. Strong, Concepts of Classical Optics, Freeman, San Francisco (1958), Appendix B. 

8. G. L. Clark, The Encyclopedia of Spectroscopy, Reinhold, New York (1960), pp. 258, 259. 

9. P. Jacquinot, "New Developments in Interference Spectroscopy,” Repts. Progr. Phys., 23, 267- 
312 (1960). 

























































Chapter 20 

SPACECRAFT THERMAL 

DESIGN 

L. H. Hemmerdinger and R. J. Hembach 

Grumman Aircraft Engineering Corporation 


CONTENTS 

20.1. Definitions. 

20.1.1. Thermal Radiation Properties Terms . . . . 

20.1.2. Space Technology Terms. 

20.2. Basic Heat-Transfer Theory. 

20.2.1. Conduction. 

20.2.2. Radiant Heat Transfer Between Two Surfaces 

20.3. Thermal Coatings for Spacecraft. 

20.3.1. Optical Design. 

20.3.2. Environmental Consideration. 

20.4. Solar System. 

20.5. Angular Dependence of Reflectivity. 

20.6. Measurement of Absorptivity and Emissivity. 

20.6.1. Spectral Measurements. 

20.6.2. Thermal Measurements. 

20.7. Spacecraft Thermal Balance. 

20.7.1. Deep Space Probes. 

20.7.2. Satellites. 

20.7.3. Thermal Design. 

20.8. Testing. 


784 

784 

785 
785 
785 
787 
787 
787 

792 

793 
793 
796 
796 
799 
811 
811 
812 
821 
823 


783 























20. Spacecraft Thermal Design 


20.1. Definitions 

The definitions given below are in accord with the notation of Worthing and Halliday 
[1]. This notation and the applied concepts have appeared in a recent report by some 
employees of NBS [21 and are rapidly being adopted throughout the aerospace indus¬ 
tries, although they are not necessarily accepted by others. These are related paren¬ 
thetically to the notation in Chapter 2. 

20.1.1. Thermal Radiation Properties Terms [1-31. 


Symbol 

Term 

Units 

Definition 

R,W . 

... Radiancy 
(Emittance) 

w cm -2 

The rate of radiant energy emission from 
a unit area of a source in all the radial 
directions of the overspreading hemi¬ 
sphere. The superscript b or 0 indi¬ 
cates a blackbody value. 

N . 

... Steradiancy 
(Radiance) 

w cm -2 

The rate of radiant energy emission per 
unit solid angle per unit of projected 
area of a source, in a stated angular 
direction from the surface. 

H . 

... Irradiance 

w cm -2 

The power per unit area incident on a 
surface. 

€. 

... Emittance 
(Emissivity) 


The ratio of the rate of radiant energy 
emission from a body, as a consequence 
of its temperature only, to the corre¬ 
sponding rate of emission from a black¬ 
body at the same temperature. 

a . 

... Absorptance 

— 

The ratio of the radiant energy absorbed 
by a body to that incident upon it. 

P . 

.... Reflectance 

— 

The ratio of the radiant energy reflected 
by a body to that incident upon it. 

e, a, p. 

.... Emissivity, 
Absorptivity, 
Reflectivity 


Special cases of emittance, absorptance, 
and reflectance; these are fundamental 
properties of a material that has an 
optically smooth surface and is suffi¬ 
ciently thick to be opaque. 

T. 

... Transmittance 


The ratio of the radiant energy trans¬ 
mitted through a body to that incident 
upon it. 


e, a , p, and r require additional qualification for precise definition. The following 
terms are used as modifiers, and the symbols are used as subscripts: A = spectral; 
T = total; H = hemispherical; N = normal. In the list below these modifiers are com¬ 
bined with emittance as an example. Each term denotes surface temperature ref¬ 
erenced to a blackbody at the same temperature. 


784 










BASIC HEAT-TRANSFER THEORY 


785 


ex. 

emittance 

Ratio of spectral radiancy (or mono¬ 
chromatic radiancy at a given wave¬ 
length) from a body to that of a 
blackbody. 

€t . 

emittance 

Total radiancy (radiation over the entire 
spectrum of emitted wavelength) from 
a body to that of a blackbody. 

€//. 

. Hemispherical 

emittance 

Radiancy from a body to that of a black¬ 
body. 

€e . 

emittance 

Steradiancy from a body to that of a 
blackbody at an angle 6. 

emittance tance when the emittance is in a 

direction normal to the surface. 

For more precision, subscripts are combined: 

e th • •• 

. Total hemispherical emittance 


€tn .... 

. Total normal emittance 


€XH .... 

. Spectral hemispherical emittance 

ex.v ••• 

. Spectral normal emittance 


20.1.2. Space Technology Terms. 



Albedo 

(Reflected Solar Radiation) 

The ratio of the total solar radiant ener¬ 
gy returned by a body to the total solar 
radiant energy incident on a body. 


Insolation 
(Solar Radiation) 

The irradiation of a body by direct total 
solar radiant energy. 


Earth 

Shine 

(Earth Radiation) 

The total radiancy from the earth as a 
consequence of its apparent tempera¬ 
ture. 


Celestial 

Sphere 

A sphere of infinite radius whose center 
is the center of the earth, and upon 
which appear projected the stars and 
other astronomical bodies. 


Ecliptic 

The great circle on the celestial sphere 
formed by its intersection with the 
plane of the earth’s orbit. 


Ecliptic 

Plane 

The plane defined by the orbit of the 
earth about the sun. 


Planet 

Inclination 

The angle between the orbit of the planet 
about the sun and the ecliptic plane. 


Equatorial 

Inclination 

The angle between the planet’s equator 
and the ecliptic. 

20.2. 

Basic Heat-Transfer Theory 



20.2.1. Conduction. The general conduction equation holds in space as in a gravity 
field, although surface interface conductances may be altered due to the effects of the 


vacuum. 











786 


SPACECRAFT THERMAL DESIGN 


P = ( 71-2 = - A(Ti - T t ) = hA(T l - T t ) 

x 


X 

where k = thermal conductivity 

Btu (hr ft °F ) _1 

w(cm °C ) -1 

x = thermal path 

ft 

(cm) 

A = cross-section area 

ft 2 

(cm 2 ) 

T = temperature 

°F 

(°K) 

k 

— = h = thermal conductance 

X 

Btu (hr ft 2 °F ) _1 

w cm - 2 (°C)- 

q,P = heat flow, power 

Btu hr 1 

w 


Thermal joint conduction in a vacuum has been analyzed by Fried [4] and by Fried and 
Costello [5], and some typical examples are presented in Fig. 20-1, 20-2, and 20-3. 



Fig. 20-1. Thermal contact conductance 
for aluminum and magnesium joints in 
vacuum [5]. 



Fig. 20-2. Thermal contact conductance 
for aluminum joint with metallic shims in 
vacuum [5]. 



Fig. 20-3. Thermal contact conductance for 
nonmetallic shims in vacuum. 





THERMAL COATINGS FOR SPACECRAFT 


787 


20.2.2. Radiant Heat Transfer Between Two Surfaces. 

P l -2 = A l F A FMT 1 4-T 2 <) 

where F A and F f are as defined in Table 20-1 [6], a- is the Stefan-Boltzmann constant, 
and the other terms are as previously defined. 


Table 20-1. Values of F a and F ( * 



f a 

F ( 

Surface Ai small compared with the 
totally enclosing surface A 2 

1 

€l 

Surfaces A t and A 2 of parallel 
discs squares, 2:1 rectangles, 
long rectangles 

Fig. 20-4 

€l € 2 

Surface dA , and parallel rectangular 
surface A 2 with one corner of 
rectangle above dA { 

Fig. 20-5 

2 

Surfaces A x or A 2 of perpendicular 
rectangles having a common side 

Fig. 20-6 

€l€2 

Surfaces A i and A 2 of infinite 
parallel planes, or surface Ai of 
a completely enclosed body is small 
compared to with A 2 

1 


Concentric spheres or infinite 
concentric cylinders with 
surfaces A i and A 2 

1 

1 

€l Al \€j / 


*For more information see [7] and [8]. 


20.3. Thermal Coatings for Spacecraft 

20.3.1. Optical Design. The thermal control of space vehicles depends ultimately 
on the radiation exchange of their surfaces with their environment. Adjustments 
depend largely upon the availability of materials with desirable spectral absorptance 
(emittance) characteristics. Materials which absorb little or no solar radiation and 
absorb (or emit) much radiation characteristic of bodies at about 300°K are very useful. 
In general, no material does this completely, but the degree can be specified by an 
a/e ratio: 


a s /e,h = 


[ H k a x dk/f H x dX 

J solar / J solar 

Rbk€\dX jf Rb\dX 

J 300 °* / J 300 °* 


where the subscript s represents solar and ^represents thermal. 

About 95.3% of the sun’s energy falls in the spectral range from 0.3 to 2.5 ix, 1.2% 
below 0.3 /x, and 3.4% above 2.5 /x. See Fig. 20-7 to 20-9 for solar and atmospheric 





788 


SPACECRAFT THERMAL DESIGN 



0 1 2 3 4 5 6 7 

„ . _ T ^ Side or Diameter 


Distance Between Planes 

Fig. 20-4. Radiation between parallel planes, directly opposed. 1-2-3-4, 
direct radiation between the planes. 5-6-7-8, planes connected by non¬ 
conducting but reradiating walls. 1-5 are discs; 2-6 are squares; 3-7 are 
2:1 rectangles; 4-8 are long narrow rectangles [9]. 



Fig. 20-5. Radiation between an element and a parallel plane. Radiation 
between surface element dA and rectangle above and parallel to it, with one 
corner of rectangle contained in normal to dA. L u L 2 = sides of rectangle; 
L 3 = distance from dA to rectangle; F A = fraction of direct radiation from dA 
intercepted by rectangle [9]. 








FACTOR 


THERMAL COATINGS FOR SPACECRAFT 


789 



DIMENSION RATIO, R 2 

Fig. 20-6. Radiation between perpendicular planes. Radiation between adjacent rectangles 
in perpendicular planes. R i is the ratio of the length of the unique side of that rectangle on 
whose area the heat-transfer equation is based to the length of the common side, y/s in figure. 
R 2 is the ratio of the length of the unique side of the other rectangle to the length of the common 
side [9]. 



Fig. 20-7. The Johnson solar spectral curve [10]. 


























790 


SPACECRAFT THERMAL DESIGN 



WAVELENGTH (/i) 

Fig. 20-8. Relative spectral distributions of albedo radiation under 
various sky conditions. 



I 


CM 

I 


S 

o 


•i 

£ 

£ 

55 

CO 


u 

J 

< 

« 

H 

U 

w 

ft 

co 



Fig. 20-9. A typical spectral emissive power curve for the thermal radiation 
leaving the earth. (The 288°K blackbody curve approximates the radiation 
from the earth’s surface, and the 218°K blackbody curve approximates the 
radiation from the atmosphere in those spectral regions where the atmosphere 
is opaque.) 


radiations. The maximum spectral energy is emitted in the 0.5 -/jl region. Therefore, 
the solar absorptance a s of a coating depends upon the thermal radiation properties of 
the coating in the 0.3 - 2.5-p, range of the spectrum. Thus 


oc s 


/•2.5m 

° 

J 0.3m 


a\Hx dX 


r 2.5 M 

I 1 

J 0.3 M 


H k d\ 


( 20 - 1 ) 
















THERMAL COATINGS FOR SPACECRAFT 791 

About 0.001% of the radiation from a 300°K blackbody falls below 2.5 fx and about 
95% falls below 40 /x. Therefore, a definition of emittance at 300°K is* 


€,h = 


r* Om 

e 

J 2.5 U 
r40fi 

J 1 

J 2.5m 


€\Rl,\ d\ 


Rbk d\ 


( 20 - 2 ) 


The 40 ~/x upper limit is arbitrary and can be altered as satellites with lower tempera¬ 
tures and longer wavelength blackbody curves are considered. The emissivities of 
some materials which have high and low ratios, and some which are also useful as flat 
absorbers and reflectors, are shown in Fig. 20-10. Characteristics of optimum and 
practical surfaces are shown in Fig. 20-11. See also Chapter 8. 



1.0 


£ 
i—* 

CO 

CO 

w 


1.0 


H 

£ 

B 

co 

W 


0 


\J 


rd J , 


Ideal 


UL 


White Paint 
/— Optimum Mirror 

Solar Reflector 
_ I _ 


2 10 

WAVELENGTH (p) 


20 


Black Paint Ideal 

1.0 

Flat Reflector 


00 

<Z) 

1 Aluminum Paint 

1 | Ideal 

Flat Absorber 

_ 1 _ 1 _ 

§ 

W 

t-Polished Silver 


10 


20 


0 2 10 

WAVELENGTH (p) 


20 


WAVELENGTH (p) 

Fig. 20-10. Representative spectral emissivity curves for four ideal surfaces. 



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 


Fig. 20-11. Radiation characteristics attainable with optimum 
surfaces compared to those attainable with currently available 
materials. 


*The ratio will often be written simply as a/e rather than a,/e r /,. 



















792 


SPACECRAFT THERMAL DESIGN 


20.3.2. Environmental Consideration. Aside from the theoretical analysis of 
thermal control systems, particular emphasis must be placed on the practical aspects 
involved in manufacture, storage, and launch conditions that have adverse effects on 
the assumed values of a s /eth. 

Extreme care must be taken to prevent scoring, fingerprints, oil films, etc., on the 
thermal coating; these will alter the ajett, ratio as prescribed by analyses. 

Ultraviolet radiation degrades the transmission of some films. A series of ultra¬ 
violet degradation values is presented in Table 20-2 [8]. 

Little work has been done on the degrading effects that meteoroid bombardment has 
on coatings, although data on the particle environment are available (Fig. 20-12). 


Table 20-2. Ultraviolet Degradation of a s 


Material Description 

Exposure ° 



Optical Data b 



Remarks 


/ 

t 

E 


Initial 


Final 



White Paints 




a. 

€ 

aje 

a* 

€ 

a„/e 


Sicon 7 x 1153 

6 

12 

72 

0.25 

0.89 

0.283 

0.33 

0.88 

0.38 

4 coats on Dow 15 on Mg HM-21A 

Sicon 7 x 1153 

6 

100 

600 

0.26 

0.90 

0.29 

0.37 

0.82 

0.45 

3 coats on Dow 15 on Mg HM-21A 

Sicon 7 x 1153 

6 

100 

600 

0.31 

0.89 

0.35 

0.59 

0.88 

0.67 

4 coats on Dow 17 on Mg HM-21A 

Skyspar A-423 

6 

25 

150 

0.23 

0.89 

0.26 

0.36 

0.91 

0.4 

4 coats on P-323-B on Dow 17 on 











Mg HM-21A 

Skyspar A-423 

6 

46 

276 

0.26 

0.93 

0.28 

0.37 

0.92 

0.4 

3 coats on P-323-B on Dow 17 on 











Mg HM-21A 

Skyspar A-423 

6 

12 

72 

0.28 

0.9 

0.31 

0.31 

0.90 

0.34 

5 coats on P-323-B on Dow 17 on 











Mg HM-21A 

Kemacryl 

6 

25 

150 

0.27 

0.80 

0.34 

0.32 

0.80 

0.4 

3 coats on Dow 17 on Mg HM-21A 

M49WC 17 

6 

25 

150 

0.26 

0.79 

0.33 

0.32 

0.80 

0.41 

4 coats on Dow 17 on Mg HM-21A 

M49WC 17 

6 

100 

600 

0.26 

0.79 

0.33 

0.33 

0.82 

0.40 

4 coats on Dow 17 on Mg HM-21A 

Fuller 517-W-l Silicone 

6 

25 

150 

0.33 

0.82 

0.40 

0.35 

0.84 

0.42 

2 mils on Mg HM-21A 

Fuller 517-W-l Silicone 

6 

25 

150 

0.28 

0.87 

0.31 

0.30 

0.85 

0.36 

4 mils on Mg HM-21A 

Fuller 517-W-l Silicone 

6 

46 

276 

0.33 

0.85 

— 

0.29 

0.87 

0.34 

3 coats on Mg HM-21A 

LMSC Research Paints 











Sodium Silicate "D” 











+ Ultrox 

6 

91 

546 

0.29 

0.82 

0.34 

0.32 

0.89 

0.37 


Sodium Silicate "D” 

+ Ultrox 

Sodium Silicate "D” 

6 

119 

714 

0.29 

0.91 

0.31 

0.29 

0.85 

0.33 


4- Ultrox 

6 

47 

282 

0.27 

0.87 

0.31 

0.47 

0.86 

0.55 


Black Paints 











Kemacryl M49 BC12 

6 

100 

600 

0.94 

0.88 

1.07 

0.92 

0.84 

1.10 

4 coats on Dow 17 on Mg HM-21A 

Micobond 

6 

105 

630 

0.94 

0.91 

1.04 

0.98 

0.87 

1.13 

4 coats on Dow 17 on Mg HM-21A 

A1 Silicone, 10043 











Fuller 171-A-152 

6 

100 

600 

0.22 

0.16 

1.34 

0.33 

0.24 

1.35 

2 mils 

Fuller 171-A-152 

6 

100 

600 

0.21 

0.16 

1.26 

0.30 

0.20 

1.48 

4 mils 

Anodized & Misc. Finishes 











Dow 15 on Mg HM-21A 

6 

12 

72 

0.23 

0.05 

4.47 

0.28 

0.05 

5.22 


Dow 15 on Mg HM-21A 

6 

105 

630 

0.17 

0.07 

2.54 

0.37 

0.34 

1.09 


Dow 15 on Mg HM-21A 
Rokide A (10546) on 

20 

12 

240 

0.18 

0.08 

2.41 

0.38 

0.11 

3.39 


aluminum 

Evaporated gold on 1/16' 

6 

95 

570 

0.31 

0.75 

0.40 

0.44 

0.74 

0.59 


Kel-F 

6 

119 

714 

0.30 

0.07 

4.05 

0.31 

0.09 

3.45 

Severe discoloration of gold with 











surface blistering of plastic. 


“ Exposure, E, (sun hours) is a product of intensity, I, (suns) and time, t, (hours). 

“Optical data include solar absorptance, infrared total hemispherical emittance, and post-exposure data. 
















ANGULAR DEPENDENCE OF REFLECTIVITY 


793 


u 

V 

03 


b£> 

O 


CO 


Q 

W 

< 

U 

M 

Q 

S 

W 

> 

O 

3 

w 

Q 

CO 

w 

2 

O 

Z 

o 

X 

o 

2 

o 

w 

Eh 

w 

S 


-4 


-6 


-8 


-10 


-12 


-14 - 


-16 - 


-18- 


-20 


Sounding Rocket, 
Doberman (273) 


- Pioneer, 
Dubin 
(276) 


Corona, 

■ Beard 

(286) p m = 3.5 


Corona 
Beard 
(286) p m = 0.05 

Selected Curve 
Far from Earth 


Sounding Rockets, 

'McChacken (274) 

Explorer 1, 

'Dubin (275, 276) 

Vanguard 3, 

LaGow and Alexander (277, 278, 279) 

Cosmic Rocket 3, 

''Nazarova (280) 

Sputnik 3, 

Cosmic Rocket 2, 

Nazarova (260) 

Whipple (154) 

Bjork (281) 

Selected Curve, 
Low-Satellite 
Altitudes 



Meteorites, 

Hawkins (289) 

Radar, 

Hawkins (292) 

Radar, 
Manning 
and Eshleman 
(254) 


Photography, 
Hawkins and 
Upton (283) 


Meteorites, 
Brown (216) 


-14 -12 


-10 


-8 -6 -4 -2 0 

METEOROID MASS (log grams) 


Fig. 20-12. Meteoric flux [11]. References on the graph are those of the 
original article. 


20.4. Solar System 

Table 20-3 contains astronomical constants which are useful in the thermal design 
of spacecraft. 

20.5. Angular Dependence of Reflectivity 

Total hemispherical emissivity and total normal emissivity for opaque surfaces 
depend upon how total directional reflectivity varies with the angle of incidence. The 
following set of equations (azimuthal symmetry case) describes the relationship of total 
hemispherical emissivity to the spectral directional reflectivity of a coating [18]. 


€th(T) 



€k(0)2tt sin 6 cos 6 dd W\ 0 ( T) d\ 



2n sin 6 cos 6 dd W^iT") dk 


(20-3) 


The total emittance of a nonblackbody at any given temperature is equal to its total 
absorptance of radiation from a blackbody at the same temperature; or 







794 


SPACECRAFT THERMAL DESIGN 


o> 



i 

u 

rC 

3 


CQ 


o q 

id o w 6 
•<* (C oot> 
t- co 

Tf< 


CQ 

CO Tf 1—< 


co i-i 
© o 


-e 


£ 

3 

o c/5 
o J 

BQ 


S! 

cu 3 

* 1 


0> 

I 


03 


o o 

r-r ^ O 

<N ^ 00 

r* o o 


00 


o 

CD 

CQ 


oo © 

CO IO 


to 

o 

CQ 

lO 

CQ 

rH 

o 

<M 

CO 

rH 

as 

00 

•<t 

CQ 

rH 

rH 




r—4 

rH 


ID 

rH 

© £ 
rH n 



a> 

o 

O 

o 

04 

o o 

tD 

<N 

o 

o 

ID 

o 

CD ID 

CQ 

CQ 

00 

ID 

o 

CD 

’'f 

t> 

<N <M 

CQ 

rH 


CD 


rf O 
t- CO 
CO rH 


rH ID 
tO -t* 


<u <u 

T3 T3 

^ o 2 


K 


o 

O 







f—4 

>—> tD 










■s 

$ 

-o 



rH 

Tf ^ 












tH 

*—* 1 —t 











o 

CD 

M 

ID CD 

tD 

rH 

o 

CD 

CQ 

CD 

CQ 



*-o 


o 


CQ CO 

t-H 

tD 

ID 

CD 

CD 

rH 

rH 





o 

o 

o o 

o 

o 

d 

O 

o 

o 

o 















CO 

H 

Z 


N 

A) 1 

o ii 
.. c ^ 


as 

CO 

■'t 

tD 


03 

CQ 

ID 

eo 

OQ 

•< 


8 3 i J? 
■2« tH « 


CO 

CD 

ci 

o 

CD 

tj* 

rH 

o 

o 

CQ 

H 



ID 

Tf 


03 

rH 





''t 

CO 

CO g 3 “ 


03 

00 


rH 






rj< 

Z 

O 

o 


(M 
























< 

o 


"3 cj 

■S q q 
k ^ • • 




03 

tD 








§ 


3 q - 




CD 

CQ 


ID 

03 




o 

c 

o 

e r 3 o" 

3 w 




CQ 

CO 

rH 

tD 

3:7 

t# 1 

CD 

tD 

t> 

03 



r-4 

o 

(3 

•*4 

HO 

0 





CQ 

CQ 


CQ 

03 

CQ 



H 

e 













CO 

< 

• o 
•^o 

CJ 

e 

o 

3 Ir 



03 

00 


rH 

ID 

o 

CQ 

CQ 


Or 

CQ 

oo 



o. 

•« q -" 


▼-H 

CO 


o 

00 

03 

CO 

cb 

CO 


• 


-O o .. 


o 

<N 


ID 


CQ 

CQ 


oq 


eo 

1 





CO 


rH 

rH 

CQ 

CD 

rH 



o 










rH 


OQ 


HO 












W 














J 






03 








03 


CO 




CO 








< 


adiu 

(km) 

[12] 

o 

o 

o 

00 

o 

o 

O 

O 

o 

00 


£h 


o 

o 

o 

l> 

rH 

00 

ID 

O 

o 

CO 




o 

tD_ 

C4 


CO 

00 

tD 

ID 

o 

t> 





CD* 

<N 

CD* 

CD 

CO 

as 

l> 

tD 

ID 

rH 





03 





CD 

ID 

CQ 

CQ 




CO 


C 



CQ 









O r 


CD 

00 

CD 

CD 

ID 

O 





q CQ 


co 

CQ 

rH 

CD 

tD 

tr 

03 

CQ 

■<* 


-c <* rl 
ft. 5- “ 


If 

o 

t> 

o 

o 

rH 

co 

rH 

■<* 

ID 

o 

o 

o 

o 

CO 

o 

CO 

03 









rH 

CQ 

CO 



c 

o 


ID 

t# 1 

CO 








*r» — 


t- 

00 

CO 

rH 

rH 

00 





rihel 

(au; 

[12] 


o 

rH 

00 

00 

ID 

o 

00 

o 

03 



CO 

o 

O'- 

o 

as 

o 

CO 

rH 

as 

o 

03 

CQ 

00 

00 

03 

CD 

03 










rH 

CQ 

CQ 


Qs 












•<*» 

s 


rH 

CO 








B 

M 

— 



CO 

o 


CO 

03 




8 

CAS ^ ^ r —1 


oo 

CQ 

o 

CQ 

o 

CO 

00 

CD 

CQ 


CO 

o 

Or 

o 

o 

rH 

ID 

rH 

CQ 

ID 

ID 

03 

rH 

03 

O 

o 

ID 

03 

J5* 








rH 

eo 

CO 

00 



& 




b> 


00 

03 

c 


00 

Body 

Sun 

3 

2 

03 

2 

Venus 

Earth 

Mars 

2 

D. 

3 

*-3 

Saturr 

3 

C 

3 

I-. 

£3 

3 

HH 

a 

03 

£ 

Pluto 

Moon 


0 ) 

-C 

s 

■8 


3 

00 

GO 

CO 


a; 

o 

co 

a 

® 


a> 

(h 

3 


In 

8 . £ 
E 3 
3 5 

| 8 
E-i -o 
O 

6 "5 

jc .3 

E 

o 

c 

2 
h* In 
as "C 

2 *> 

^ c 


& 


3 

T3 

'o 

C 

4) 

O 

c 

-*-> 

3 

o 

• m 
> 
ai 
u, 


C 

o 


5*8 

< o 

w 

3 3 

3 C 
_ be 

«8 S 

E 1 

o « 

o £ 

3 E 

a> 3 
« 2 
rH O 


- 'O 
C as 

O t- 

■•§ lg 

"i « 

+2 c 
c co 

<D 0) 

«! 

■«_> 00 

#? 




ANGULAR DEPENDENCE OF REFLECTIVITY 


795 


€ T h(T) = octh(T) (Kirchhoff’s law) (20-4) 

where €x(0) = specular spectral emissivity 

£th(T) = total hemispherical at temperature T in degrees absolute 

&th(T) = total hemispherical absorptivity for the same source temperature T in 
degrees absolute 

W\°{T) = Rb\ (T) = Planck’s blackbody function 
Ai, X 2 = wavelength limits to cover 95% of Rb(T) 


The equation for total normal emissivity which is readily measured in the laboratory, 
would be 


Ct\{T) 


j: 


€k(0)Wk°(T) d\ 


r \ 2 

W K » 

J\, 


6 = 0 ° 


(20-5) 


(T) dk 


€t\(T) = 1 — ptn(T) (for opaque surface) (20-6) 

€tn(T) = oc T n(T) (Kirchhoff’s law) (20-7) 

For metal surfaces the ratio €thI€tn is greater than 1 and for dielectric surfaces may 
vary from 0.92 to slightly above 1. The relation is given in Fig. 20-13 as a function 
of the measurable normal emissivity [8]. This is an empirical relation. For an ideal 
diffuse emitter (according to Lambert’s law), €th/€t:\ = 1. 



Fig. 20-13. Ratio of total hemispherical emissivity to 
normal emissivity. The curve is from Jakob [8]. 





796 


SPACECRAFT THERMAL DESIGN 


20.6. Measurement of Absorptivity and Emissivity 

Two distinct measurement techniques are employed for obtaining a s and €th, the 
spectral reflection technique and the thermal techniques. 

20.6.1. Spectral Measurements. The spectral reflection method is based on the 
definition of reflectivity which requires that the reflected energy be measured over the 
entire overspreading hemisphere. In determining a 8 , the definition is complied with 
in the 0.25-2.5-/x range using an integrating sphere-spectrophotometric apparatus [19]. 
The resultant property measured is the directional spectral reflectivity. Integration 
of the spectral reflectivity data over the Johnson curve (by IBM programming or equal- 
energy increment-summation techniques) yields total normal solar reflectivity. When 
this solar reflectivity is subtracted from one, a s is realized. 

On the basis of the reflectivity-reciprocity relation, the normal or directional spectral 
reflectivity in the 2.5-50-//. range may be measured with a heated-cavity reflectometer- 
spectrophotometer type of apparatus [20]. The sample is irradiated hemispherically 
and viewed at a specific angle by the spectrophotometer’s optical system. Integra¬ 
tion of the spectral reflectivity data over the appropriate blackbody curve characteristics 
of the spacecraft temperature gives the total normal emissivity properties of the mate¬ 
rial. The total normal emissivity must be converted to total hemispherical emissivity 
(use Fig. 20-12). For precise thermal-control design, the normal spectral reflec¬ 
tivity data should be supplemented with directional reflectivity information and an 
integration (or discrete summation) performed. The normal spectral reflectivity data 
for several materials are shown in Fig. 20-14 to 20-19 [21]. 



Fig. 20-14. Compiled spectral reflectivities. A = evaporated aluminum, 25 /xin. on polished 
6061-T6 aluminum. B = evaporated aluminum (0.2 /x), on 1/4-mil Mylar crumpled and stretched, 
(looking at aluminum). C = evaporated aluminum (0.2 /a) on 1/4-mil Mylar crumpled and stretched 
(looking at Mylar). 
























MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 


797 



Fig. 20-15. Compiled spectral reflectivities. A = chromic acid anodize on 24S-T81 aluminum. 
B = sulfuric acid anodize on 24S-T81 extruded aluminum, chemically milled. C = hard anodize 
(1 mil) on 6061-T6 aluminum (35 amp ft -2 , 45 volts, 20°F). 



WAVELENGTH (m) 


Fig. 20-16. Compiled spectral reflectivities. A = polished copper, 17 mils thick. B = Tabor 
solar collector chemical treatment (110-30) on nickel-plated copper. C = Tabor solar collector 
chemical treatment (125-30) on nickel-plated copper. 
















798 


SPACECRAFT THERMAL DESIGN 



WAVELENGTH (m) 

Fig. 20-17. Compiled spectral reflectivities. A — immersion gold, approximately 0.03 mil, on 
nickel-plated copper plate on 6061-T6 polished aluminum, aged 6 months in air, unpolished. 
B = vacuum-evaporated gold on fiberglass laminate. C = gold ash (80 fi) on 0.4-mil silver on 
Epon glass. D = white gold on polished MIL-S-5059 steel. 



WAVELENGTH (fi) 

Fig. 20-18. Compiled spectral reflectivities. A = silicon solar cell, International Rectifier 
Corp. B = silicon solar cell, International Rectifier Corp. 1.11-/* evaporated coat, SiO —fast 
deposition rate. C = silicon solar cell, Hoffman Corp. Type 120-C, 3-mil glass cover. 


cqcuQ 
































MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 


799 



Fig. 20-19. Compiled spectral reflectivities. A = Flat white paint, Fuller No. 2882 on 2-mil 
polished aluminum. B = white epoxy resin paint of Mg. C = flat white acrylic resin, Sherwin- 
Williams M49WC8-CA-10144; MIL-C-15328A pretreatment wash coating on 22-mil 301 stain¬ 
less steel 1/2 hard. D = white paint mixed with powdered glass, 7 mil on polished aluminum. 


20.6.2. Thermal Measurements. Thermal measurement of a s and e t h will yield €th 
directly, but the a s measurement’s validity depends on matching the solar simulator 
source with the spectral distribution of the correct spectrum. A normalized spectral 
curve of the solar carbon arc and a bare xenon lamp shows that the carbon arc simulates 
the solar spectrum more closely than other light sources. Measurements can be 
equilibrium or dynamic. 

Equilibrium Measurements [18, 211. With electrical heating and the property €ts, 
the governing relation is 


I 2 R 


6th — 


o-(T 2 4 - 7V) 


With a solar simulator as the heater and the property a s /eTH, it is 


£th 


crA s 

WAp 


(T 2 4 - TV) 


Dynamic Measurements [18]. With the solar simulator as the heater and the 
property a s /e T H, one makes cooling and heating measurements. For heating, the 
relationship is 


mc p 


dT s 

dt 


= A p a s W + P — A s €th &TV 


mcp 


dT s 

dt 


= P' 


As€tH<tT s 4 


For cooling, it is 




















SPACECRAFT THERMAL DESIGN 


800 

In these equations, 

/ = current to sample heater 
R = resistance of sample heater 
T 2 = equilibrium temperature of sample 
T i = temperature of cold wall of vacuum chamber 
T s = sample temperature 
A s = total surface area of sample 

A p = projected area of sample as viewed in the direction of illumination 
W — total radiant power per unit area from solar simulator 
m = mass of sample 

c p = specific heat of sample per unit mass 

P = incident thermal radiation power from walls of vacuum chamber 
P' = incident thermal radiation for the cooldown period 
t = time 

Since the thermal radiation properties are dependent not only on the intrinsic mate¬ 
rial but also on the coatings, thickness, and surface condition, published numerical 
values for a s and e t h must be used cautiously. Data on the values of a s and e t h should 
include a complete statement about the coating: surface roughness, thickness, cleanli¬ 
ness, precise chemical composition, and a complete substrate description [22]. Current 
published data seldom supply sufficient descriptions of coatings, may even fail to dis¬ 
tinguish between e™ and e T h, or may not associate a temperature with a given emittance 
figure. The thermal radiation properties data tabulation (Table 20-4) is meant only to 
be a guide in materials selection. After a coating has been selected and its method 
of formulation, application, and handling fixed, the a s and e,h properties should be 
measured. 

In Fig. 20-20 to 20-28, the emissivities of materials and their solar absorptances 
are given. More details on measurement conditions, etc., are given in [12]. 



Fig. 20-20. Normal total emissivity and total solar absorp¬ 
tivity of Inconel X. (Note: temperatures are those to which 
samples had been heated before tests.) 



MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 


801 


0.5|— 


0.3 - 


>* 

es 

w 
OT 

t—i 

s 

w 

< 

H 

O 

H 

a 0.2 

s 

w 

33 

d< 

1 0.. 

W 

S3 


0 


0.4 - -o_ 


Polished 




10-mil-Diameter Wire 


Sandblasted 


■vwr 


Polished 


Vapor blasted 
Polished 

Chemically Cleaned 

„r'' 


± 




0.5 


0.4 K 

£ 

H 

d< 

03 

g 

3 

03 

< 

o 

co 

•J 
< 
H 
O 
H 


0.3 


0.2 


0.1 


1000 


4000 


0 

5 000 


2000 3000 

TEMPERATURE (°F) 

Fig. 20-21. Hemispheric total emissivity and total solar absorptivity of molybdenum. 



H 

£ 

H 

(X 

03 

8 

3 

03 

< 

•J 

O 

(Z) 

< 

Eh 

O 

Eh 


Fig. 20-22. Normal total emissivity and total solar absorptivity for K-Monel. 
temperatures are those to which samples had been heated before tests.) 


(Note: 



H 

»—I 

Eh 

A 

03 

O 

OT 


03 

< 

O 

<Z) 

tJ 

< 

H 

O 

Eh 


Fig. 20-23. Normal total emissivity and total solar absorptivity for 301 
stainless steel. (Note: All measurements made at 100° F. Temperatures 
are those to which samples had been heated previous to tests.) 










802 


SPACECRAFT THERMAL DESIGN 


H 


co 

CO 


s 


w 

< 

H 

O 

H 

-J 


£ 

O 

2 


1.0 


0.8 - 


0.6 - 


0.4 


0.2 


0.0 


■400 


o Polished 
a Clean and Smooth 
® As Received 




/ o- 


0 

. O-’, / 

»•* / / 

/ 


o- 


O 

6 A 


_L 


_L 


I 


I 


400 800 1200 1600 

TEMPERATURE (°F) 


1.0 


0.8 


0.6 


0.4 


0.2 


0.0 


2000 2400 


>* 

H 

I—I 

£ 

H 

(X 

£ 

o 

co 

ffl 

£ 

< 

o 

co 

J 

c 

O 

H 


Fig. 20-24. Normal total emissivity and total solar absorptivity for 316 
stainless steel. (Note: All measurements made at 100° F. Temperatures 
are those to which samples had been heated previous to tests.) 



TEMPERATURE (°F) 


H 


£ 

£ 

O 

co 


03 

< 

£ 


O 


co 


< 

E- 

O 

H 


Fig. 20-25. Normal total emissivity and total solar absorptivity for 347 
stainless steel. (Note: All measurements made at 100° F. Temperatures 
are those to which samples had been heated previous to tests.) 



Fig. 20-26. Normal total emissivity for 18-8 stainless steel. 









HEMISPHERICAL TOTAL EMISSIVITY 


MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 


803 



Fig. 20-27. Hemispherical total emissivity, normal total emissivity, 
and total solar absorptivity for tantalum. 



Fig. 20-28. Hemispherical total emissivity of tungsten. 


NORMAL TOTAL EMISSIVITY 












804 


SPACECRAFT THERMAL DESIGN 


Table 20.4. Thermal Radiation Properties of Materials 


Material 

a. 

Temperature 

(°R) 

€r.v 

€th 

a ,/cth 

Ref. 

Aluminum 







Commercial plate 


671 

.09 



23 

Commercial plate, polished 


671 

.05 



23 

Commercial plate dipped in HN0 3 
Commercial plate dipped in hot 


671 

.05 



23 

hydroxide 

2024 aluminum alloy 


671 

.04 



23 

as received 

0.27 

400 


0.02 

13.5 

24 



500 


0.02 

13.5 




600 


0.02 

13.5 


2024 aluminum alloy cleaned 

0.34 

400 


0.06 

5.666 

24 



500 


0.06 

5.666 




600 


0.07 

4.857 


2024 aluminum alloy mechan¬ 







ically polished and degreased 

0.31 

400 


0.05 

6.20 

24 



500 


0.06 

6.20 




600 


0.06 

5.16 


2024 aluminum alloy sand 







blasted 

0.67 

400 


0.25 

2.68 

24 



500 


0.27 

2.48 




600 


0.30 

2.23 


6061 aluminum alloy as received 

0.41 

400 


0.05 

8.2 

24 



560 


0.04 

10.25 


6061 aluminum alloy chemically 







cleaned 

0.18-0.44 

400 


0.03-0.11 


24 



500 


0.03-0.12 





600 


0.04-0.12 



6061 aluminum alloy polished 







and degreased 

0.35 

400 


0.04 

8.75 

24 



500 


0.04 

8.75 




560 


0.05 

7.00 


6061 aluminum alloy 120-size 







grit blasted 

0.60 

400 


0.40 

1.5 

24 



500 


0.41 

1.46 




600 


0.41 

1.46 


Alzak on aluminum (190 /iin. 







thick) 

0.15 

410 

0.79 



21 



530 

0.77 






580 

0.75 




Aluminum foil Reynolds wrap 







shiny side as received 

0.19 

400 


0.03 

6.333 

24 



500 


0.04 

4.75 


Aluminum foil Fas son adhesive- 







backed 

0.17 

400 


0.03 

5.66 

24 



500 


0.03 

5.66 


Mylar metalized with vacuum- 
deposited aluminum 

0.20 

400 


0.05 

4.0 

24 



500 


0.05 

4.0 


Aluminum vacuum deposited on 







magnesium with standard 
silicone undercoat 

0.13 

400 


0.04 

3.25 

24 



500 


0.04 

3.25 




600 


0.04 

3.25 


6061 T-6 aluminum hard anodize 







1 mil thick 

0.923 

450 


0.841 

1.10 

21 



410 

0.830 



23 


460 0.842 

560 0.863 


MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 805 


Table 20-4. Thermal Radiation Properties of Materials ( Continued) 


Material 

a* 

Temperature 

(°R) 

€tn 

*TH 

CXs/iTH 

Ref. 

Antimony 







Polished 


671 

0.03 



23 

Rolled plate 


540 

0.06 



23 

Shim stock 6 5/35 


531 


0.029 


23 

Oxidized 


900 

0.60 



23 



671 

0.60 




Beryllium 







QMV Beryllium alloy 64 RMS 







finish 

0.70 

400 


0.16 

4.375 

24 



600 


0.17 

4.117 


Cadmium 


540 

0.02 



23 

Electroplate (mossy) 


531 


0.03 


23 

Chromium 


540 

0.08 



23 

Chromium plate 0.1 mil thick on 







0.5 mil nickel plate on 321 
stainless steel exposed to JP-4 
combustion products 50 hours 
at 1,100°F 

0.778 

555 


0.15 

5.18 

21 

Cobalt 


531 

0.03 



23 

Columbium (niobium) 






26 

Columbium alloy (Cb-lOTi-lOMo) 






26 

Copper 







Black oxidized 


540 

0.78 



23 

Scraped 


540 

0.07 



23 

Commercial polish 


540 

0.03 



23 

Electrolytic, careful polish 


635 

0.018 



23 



531 


0.015 



Chromic acid dip 


531 


0.017 


23 

Polished 


531 


0.019 


23 

Liquid honed 


531 


0.088 


23 

Electrolytic polish 


431 

0.006 



23 

Mechanical polish 

Pure copper, carefully prepared 


531 

0.015 



23 

surface 


531 

0.008 



23 

Ebanol 







Ebanol C on copper treated 5 







min at 196° F in 219°F boiling 
point solution 

0.908 

555 


0.11 

8.25 

21 

Ebanol S on steel treated 15 







min in a 286°F boiling solution 

0.848 

555 


0.10 

8.49 

21 

Glass 3 mils thick on silicone 







solar cell 

0.925 

450 


0.843 

1.10 

21 

Gold 







0.000010-in. leaf (on glass or 

Lucite plastic) 

0.000040-in. foil (on glass or 


531 


0.063 


23 

Lucite plastic) 

0.0005-in. foil (on glass or 


531 


0.023 


23 

Lucite plastic) 

0.0015-in. foil (on glass or 


531 


0.016 


23 

Lucite plastic) 

Gold plate 0.00005 in. on stain¬ 


531 


0.01 


23 

less steel (1% Ag in Au) 


531 


0.027 


23 


806 


SPACECRAFT THERMAL DESIGN 


Table 20-4. Thermal Radiation Properties of Materials ( Continued ) 


Material a s 

Gold (Continued) 

Gold plate 0.0001 in. on stain¬ 
less steel (1% Ag in Au) 

Gold plate 0.0002 in. on stain¬ 
less steel (1% Ag in Au) 

Gold plate 0.0002 in. on copper 
(1% Ag in Au) 

Gold vaporized onto 2 sides of 
0.0005-in. Mylar plastic 
Deep electroplated gold on 
aluminum 


Gold plate on 7075 aluminum 

0.30 

Vacuum-deposited gold on buffed 
titanium 

0.33 

Vacuum-deposited gold on alumi¬ 
num with resin undercoat 

0.24 

Graphite (crushed carbon electrodes) 
16 mils thick on sodium 
silicate on polished aluminum 

0.96 

Inconel 

Inconel foil (0.005) as received 

0.55 

Inconel X (Fig. 20-20) 

Inconel X oxidized 4 hr at 

1825°F in air followed by 10 hr 
at 1300°F in air 

0.898 


Iridium 

Iron 

Electrolytic 

Cast iron, polished 
Cast iron, oxidized 

Iron sheet, rusted red 
Iron, oxidized 

Iron, nickel alloys 
Tinned iron sheet 
Galvanized iron 

Lead 

Unoxidized, polished 
Gray oxidized 
Oxidized at 473°K 
Red lead 

Lead foil 0.004 in. 
Magnesium 


Temperature 

(°R) 

€tn 

€tH 


Ref. 

531 


0.027 


23 

531 


0.025 


23 

531 


0.025 


23 

531 


0.02 


23 

410 

0.02 



27 

460 

0.02 




560 

0.02 




400 


0.03 

10.0 

24 

500 


0.03 

10.0 


400 


0.05 

6.6 

24 

500 


0.05 

6.6 


600 


0.05 

6.6 


400 


0.04 

6.0 

24 

500 


0.04 

6.0 


600 


0.04 

6.0 



450 

0.908 

1.06 

21 




26 

400 

0.21 

2.619 

24 

500 

0.23 

2.391 

26 


450 


0.711 

1.26 

21 

531 

0.04 



23 





26 

959 

0.07 



23 

671 

0.05 




531 

0.05 




560 

0.21 



23 

560 

0.63 



23 

959 

0.66 



23 

1460 

0.76 




531 

0.69 



23 

2700 

0.89 



23 

671 

0.74 



26 

535 

0.064 



23 

657 

0.07 



23 

671 

0.05 



23 

531 

0.28 



23 

851 

0.63 



23 

671 

0.93 



23 

531 


0.036 


23 

531 

0.07 



23 

959 

0.13 




1460 

0.18 





MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 807 

Table 20 - 4 . Thermal Radiation Properties of Materials ( Continued ) 


Material 

Manganin, bright rolled 
Molybdenum 


Molybdenum as received (Fig. 20-21) 0.48 
Monel 

K Monel 5700 (Fig. 20-22) 

Nickel 
Polished 
Bright matte 
Nickel foil 0.004 in. 

Electrolytic 


Electroplated 
on iron and unpolished 
on iron and polished 
on copper 

Electroless nickel (Dow 
Chemical Co.) 0.45 

Oxidized 

Nichrome 

"Driver Harris” nichrome 
heater strip 

Paints 

Mixing white lacquer, 

4 coats, 2 coats zinc chromate 

primer, thickness total 0.006 in. 0.21 

(Ditzler) 

Sherwin-Williams Kemacryl 
white paint No. M49WC12, 4 
mils over one wash primer coat 
on alodined 2024-T6 aluminum 


Kemacryl lacquer white No. 
M49WC17 4 wet coats, 1 coat 
pretreatment primer P40GC1 
(Sherwin-Williams) 0.26 


Zinc sulfide pigment, acryloid 
A-10 binder, pigment to volume 
concentration 30%, 5.5 mils over 
one wash primer coat on alo¬ 
dined 2024-T6 aluminum 


Temperature 

(°R) 

€tk 

*TH 

<*»/€th 

Ref. 

704 

0.048 



23 

531 

0.076 




4140 

0.24 



23 

3240 

0.19 



26 

2341 

0.13 




671 

0.07 




531 

0.05 




500 

0.12 


4.0 

24 





26 

711 

0.045 



26 

23 

711 

0.041 



23 

531 


0.022 


23 

1460 

0.10 



23 

959 

0.07 




560 

0.06 




531 

0.04 




531 

0.11 



23 

531 

0.045 



23 

540 


0.03 


23 

400 


0.16 

2.815 

24 

500 


0.17 

2.647 


2700 

0.85 



23 

900 

0.37 





460 

0.11 

27 

560 

0.12 


660 

0.13 



400 

0.85 

0.247 24 

500 

0.85 

0.247 

600 

0.86 

0.244 


410 

0.90 

27 

460 

0.91 


560 

0.91 



400 

0.73 

0.356 24 

500 

0.75 

0.346 

600 

0.77 

0.337 


410 

0.90 

27 

460 

0.90 


560 

0.90 



808 


SPACECRAFT THERMAL DESIGN 


Table 20 - 4 . Thermal Radiation Properties of Materials ( Continued ) 


,, . , Temperature . D , 

Material a * , ot> , e™ *th ocs/*th ttef. 

( K) 

Paints ( Continued) 

Lithofrax 72662, 10 mils thick on 


vapor honed titanium substrate 

410 

0.92 


27 


460 

0.93 




560 

0.93 



Lithofrax 72662, 11 mils on vapor 





honed 2014-T6 substrate 

410 

0.92 


27 


460 

0.92 




560 

0.93 



White paint, sodium silicate 





binder, zinc oxide-zinc sulfide 
pigmented, substrate vapor 
honed 2024-T3 

410 

0.94 


27 


460 

0.94 




560 

0.95 



Zinc oxide pigment, spectral grade 





in potassium silicate binder 
thickness 5 mils, substrate vapor 
honed 6061-T4 aluminum alloy 

410 

0.94 


27 


460 

0.95 




560 

0.95 



White epoxy resin paint 





"Cat-a-lac” Finch Paint and 

Chemical Co. No. 463-1-8 on 
aluminum 0.248 

450 


0.882 

0.28 21 


Skyspar A-423 color SA8818 
(white) 4 wet coats over one 
coat epoxy primer P-323 
(Andrew Paint Co.) 0.22 


517-B-2 white gloss 
silicone (W. P. Fuller Co.) 

3 wet coats 0.30 


Zinc sulfide pigment in DO 808 
silicone vehicle, 5-1/2 mils 
thick substrate vapor honed 
2014-T6 aluminum alloy 

Pemco No. R46H60 porcelain 
enamel on steel 0.22 


Strong black lacquer, 

4 coats, 2 coats zinc chromate 

primer, thickness .002 in. 0.92 

(Ditzler) 


Kemacryl lacquer black No. 
M49BC12, 4 wet coats, 1 coat pre¬ 
treatment P40GC1 (Sherwin- 
Williams) 


400 

0.82 

0.268 24 

500 

0.83 

0.265 

600 

0.84 

0.261 


400 

0.82 

0.365 24 

500 

0.81 

0.370 

600 

0.80 

0.375 


410 

0.90 



27 

460 

0.90 




560 

0.90 




400 


0.76 

0.289 

24 

500 


0.78 

0.282 


600 


0.79 

0.278 



400 

0.71 

1.296 24 

500 

0.74 

1.243 

600 

0.75 

1.227 


400 

0.81 

1.160 

500 

0.83 

1.133 

600 

0.84 

1.119 


0.94 


24 


MEASUREMENT OF ABSORPTIVITY AND EMISSIVITY 


809 


Table 20 - 4 . Thermal Radiation Properties of Materials ( Continued ) 


Material 

Paints ( Continued) 

Black epoxy paint 1019 
2 wet coats over 1 coat 1012 
primer on alodined aluminum 
2014-T6 (Grumann) 


Flat black epoxy resin paint 
"Cat-a-lac” Finch Paint and 
Chemical Co. No. 463-1-8 on 


a* 


T emperature 

(°R) 




€tH 


ats/tTH Ref. 


410 

460 

560 


0.90 

0.90 

0.91 


27 


aluminum 

Cat-a-lac black epoxy paint 

0.951 

450 


0.888 

1.07 

21 

2 coats on 300 series stain¬ 
less steel 


410 

0.89 



27 



460 

0.89 






560 

0.90 




Black epoxy paint applied 







on 2014-T6 alodined aluminum 


410 

0.91 



27 

(Wamow) 


460 

0.92 






560 

0.92 




517-B-2 flat black silicone 







4 mils wet film (Fuller) 

0.89 

400 


0.80 

1.113 

24 



500 


0.81 

1.099 




600 


0.82 

1.085 


3M velvet black paint over zinc 
chromate primer on alodined 
aluminum substrate 


410 

0.92 



27 



460 

0.92 






560 

0.93 




Krylon 1602 black paint over 







aluminum 


410 

0.86 



27 



460 

0.86 






560 

0.88 




Aluminum leafing pigment in 
clear epoxy binder 







Oz. pigment/gal 







10 


560 

0.59 



27 

20 


560 

0.50 




40 


560 

0.34 




50 


560 

0.35 




80 


560 

0.45 




Palladium 


531 

0.03 



23 

Platinum 


2461 

0.18 



23 



1460 

0.10 






959 

0.06 






671 

0.05 






531 

0.03 






522 

0.016 




Quartz (fused) 


531 

0.932 



23 

Rhodium, plated on stainless steel 

Rokide, flame-sprayed alumina on 


531 

0.05 



23 

410 stainless steel heated to 







1300° F in air in 60 sec and 
held 30 sec more 

0.276 

450 


0.801 

0.34 

21 


810 


SPACECRAFT THERMAL DESIGN 


Table 20 - 4 . Thermal Radiation Properties of Materials ( Continued ) 

,, . , Temperature . „ . 

Material a, e™ *th a s /e T H Ref. 

I k; 

Silicon Solar Cell (International 
Rectifier Corp.), approximately 
1 mm thick on electroless nickel 


plate substrate, boron-doped 


surface 

0.938 

555 


0.316 

2.97 

21 

Silver 


1460 

0.03 



23 



671 

0.025 






531 

0.022 






492 

0.02 




Solder, 50-50 on Cu 


531 


0.032 


23 

Stainless steel 







Polished 


671 

0.08 



23 

Type 301 (Fig. 20-23) 






26 

Type 302 


760 


0.048 


23 

Type 302 mechanically polished 

0.38 

400 


0.17 

2.24 

24 



500 


0.19 

2.0 




600 


0.20 

1.90 


Type 316 (Fig. 20-24) 






21 

Type 347 (Fig. 20-25) 






21 

Type 410 heated to 1300°F in air 

0.764 

555 


0.130 

5.88 

21 

Type 18-8 (Fig. 20-26) 






21 

Armco black oxide on type 301 

Rene 41 alloy mechanically 

0.891 

450 


0.746 

1.19 

21 

polished 

0.38 

400 


0.17 

2.24 

24 



500 


0.19 

2.0 




600 


0.20 

1.90 


Tabor 







Tabor solar collector chemical 
treatment of galvanized iron 

Tabor solar collector chemical 

0.885 

555 


0.122 

7.25 

21 

treatment 110-30 on nickel 
plated copper 

0.853 

555 


0.049 

17.4 

21 

Tantalum (Fig. 20-27) 






26 



4140 

0.26 



23 



3240 

0.21 






531 

0.05 




Tellurium 


531 

0.22 



23 

Titanium 







Alloy Ti-6AL-4VA 






26 

Alloy Ti-6AL-4VA as received 

0.66 

400 


0.19 

3.47 

24 



500 


0.20 

3.3 




600 


0.22 

3.0 


Titanium vapor coated on bright 







side of Reynolds wrap alumi¬ 
num foil, 80 to 100 p thick, 
heated 3 hr at 750°F in air 

0.746 

555 


0.138 

5.40 

21 

Titanium C-110M (AMS 4908) 
heated 300 hr at 850° F in air 
Titanium C-110M (AMS 4908) 

0.768 

555 


0.198 

3.88 

21 

heated 100 hr at 800° F in air 
Titanium 75A (AMS 4901) heated 

0.524 

555 


0.162 

3.24 

21 

300 hr at 850° F in air 

0.798 

555 


0.211 

3.78 

21 

Anodized titanium 

0.515 

450 


0.866 

0.59 

21 

Titanox-RA 2 mils thick on 







black paint 

0.154 

450 


0.885 

0.17 

21 


SPACECRAFT THERMAL BALANCE 


811 


Table 20-4. Thermal Radiation Properties of Materials ( Continued ) 


Material 

Temperature 

(°R) 

*TN 


a,/€ T H Ref. 

Tin 




1% indium 

531 

0.012 


23 

5% indium 

531 

0.017 


23 

Tin, 0.001 in. foil 

531 

0.012 


23 

Tungsten (Fig. 20-28) • 




26 

Filament 

4140 

0.28 


23 


3240 

0.23 




2340 

0.15 




1440 

0.088 




900 

0.053 




540 

0.032 



Zinc 

531 

0.05 


23 


20.7. Spacecraft Thermal Balance 

20.7.1. Deep Space Probes. Spacecraft traveling between planets are thermally 
influenced by the solar heat flux. In Fig. 20-29(a) the solar heat flux associated with 
planet distance from the sun is plotted as the ordinate [13]. The surface temperatures 
of an insulated flat plate normal to the sun are plotted as functions of its coating a s /e,h 
and its coordinate within the solar system. 


o 


Z 

l=> 

CO 

2 

o 

K 

fa 

fa 

u 


a 

Q 



Fig. 20-29a. Solar heat flux as a function of distance and a/e. 










812 


SPACECRAFT THERMAL DESIGN 


The heat flux (power) to any surface incremental area at an arbitrary position in 
space is computed by multiplying the solar power at the particular probe distance from 
the sun by the cosine between the surface normal and the sun line. For an average 
value over the spacecraft surface, this is the projected surface area to the sun. 

Thus if the solar constant is written S, then the solar irradiance is: 

H s = S cos 0 (20-8) 

Habsorbed H s OC s (20-9) 


For a perfectly insulated unit area surface, the temperature, T s f c , is: 



The temperature of space (= 4°R) is assumed negligible for ordinary spacecraft surface 
temperatures. 

20.7.2. Satellites. A satellite, i.e., a spacecraft orbiting a planet, is irradiated from 
three external sources: 

(1) The sun (solar radiation) 

(2) Reflected sunlight off the planet (albedo radiation) 

(3) The planet surface acting as a blackbody (planet radiation) 



Fig. 20-296. Percentage of time in sun vs . altitude with orbit 
inclination /3 as parameter for vehicle in circular orbit. 


















SPACECRAFT THERMAL BALANCE 


813 


20.7.2.1. Incident Solar Radiation. The solar radiation power in a planet’s orbit 
is obtained from Table 20-3. 

In orbit, a satellite in the shadow of a planet is occulted from the sun. This shad¬ 
owing, and time in the sunlight, varies according to the angle (3 between the satellite’s 
orbital plane and the sun-planet plane (the ecliptic, when considering the earth; see 
Fig. 20-296). The angle /3 is determined by the incident launch angle i. The orbit of 
an earth satellite precesses; i.e., the normal to the orbital plane will generate a cone 
about the earth’s axis at a constant half-angle of i. A 500-mi circular orbit at i = 32° 
precesses approximately 6° per day. Because of the precession of the orbit and the 
rotation of the earth about the sun, the angle f3 varies continuously from a maximum 
(equatorial inclination +i) to a minimum (equatorial inclination —i). The maximum 
and minimum average solar heat load over the orbit are determined, respectively, 
when (3 is a maximum and when (3 = 0°. 

Consequently, the orbital average solar heat flux incident at a point on a spatially 
oriented satellite’s surface is 

H s = S cos 6 x (% time in sun) (20-10) 

where 9 is the angle between the surface normal and the sun-line. 

20.7.2.2. Incident Albedo Radiation. The parameters in the orbit geometry (Fig. 
20-30) used in defining the incident albedo flux are: 

9 S = angle between the planet-sun line and the planet radius vector to the satellite 

y = angle between the satellite surface normal and the planet radius vector to the 
satellite surface 

v = orbit angle about the planet 

<f> c = one of the attitude parameters, the angle of rotation about the planet radius 
vector to the surface normal 

<f) c = 0 when the normal lies in the plane containing the planet-surface vector and 
the planet-sun vector 

c f ) c , dg, and y may vary as the satellite traverses its orbit; therefore, to obtain average 
albedo heat fluxes, computations of geometric factors must be made at each orbit angle v 
and integrated over the orbit. 

Figures 20-31 through 20-41 are used to determine the geometric factor F a for albedo 
radiation incident to a sphere, cylinder, and flat plate [13]. 


Sun 



Fig. 20-30. Sun-vehicle geometry. 


GEOMETRIC FACTOR 


SPACECRAFT THERMAL DESIGN 


814 



Fig. 20-31. Geometric factor for planetary albedo radiation 
incident to a sphere, versus altitude, with angle of sun as a 
parameter. 



Fig. 20-32. Geometric factor for albedo to hemisphere, versus 
altitude, with angle of sun as parameter (y = 0°, <t> c = 0°). 



















SPACECRAFT THERMAL BALANCE 


815 



ALTITUDE Planet Radii 


u l) 


Fig. 20-33. Geometric factor for albedo to hemisphere, versus 
altitude, with angle of sun as parameter {y = 90°, <f> c = 0°). 



ALTITUDE Planet Radii 


11 1 ) 


Fig. 20-34. Geometric factor albedo to cylinder, versus altitude, 
with angle of sun as parameter (y = 0°, <J) r = 0°). 













GEOMETRIC FACTOR 


816 


SPACECRAFT THERMAL DESIGN 




10 


ALTITUDE 'Planet Radii 




Fig. 20-35. Geometric factor for albedo to cylinder, versus 
altitude, with angle of sun as parameter (y = 90°, <t> c = 0°). 



ALTITUDE 


(Planet Radii 




Fig. 20-36. Geometric factor for albedo to one side of flat plate, 
versus altitude, with angle of sun as parameter (y = 90°, <t> c = 0°). 





















SPACECRAFT THERMAL BALANCE 


817 



Fig. 20-37. Geometric factor for albedo to one side of flat plate, 
versus altitude, with angle of sun as parameter (y — 0°, <b c = 0°). 



Fig. 20-38. Geometric factor for planetary thermal radiation 
incident to sphere, versus altitude. 


















818 


SPACECRAFT THERMAL DESIGN 



io" 2 lO -1 1 10 

ALTITUDE ^Planet Radii 


Fig. 20-39. Geometric factor for planetary thermal radiation 
incident to hemisphere, versus altitude, with attitude angle as 
parameter. 



ALTITUDE ^Planet Radii | 


Fig. 20-40. Geometric factor for planetary thermal radiation to 
cylinder, versus altitude, with attitude angle as parameter. 

















SPACECRAFT THERMAL BALANCE 


819 



Fig. 20-41. Geometric factor for planetary thermal radiation 
incident to flat plate, versus altitude, with attitude angle as 
parameter. 


From these, the instantaneous albedo heat flux 
is obtained: 


H' a = SaF a 


density indicated by the prime 


where a is the albedo factor (from Table 20-3). 
Average heat fluxes per orbit are obtained from 


n=k 

Ha —Sa 2 Fa 

n= 1 

V H 

where v n = 0°, 10°, 20°,...350° 
n= 1, 2, 3,...k 

To obtain the total power incident, P = HA, one can project the surface area for 
A in Fig. 20-42 through 20-45. For the flat-plate heat fluxes, the cosine relation is 
incorporated in the curves, so that the full flat-plate area can be used. 

20.7.2.3. Incident Planetary Radiation. Similar geometric factors [13] are given 
below for planetary radiation to a sphere, cylinder, and flat plate using the geometry 
outlined above. The angle y varies with orbit angle, and for average orbital planet 
flux the geometric factor must be averaged over the orbit. 



Generally the planet emissivity e p is assumed to be 1. T p is obtained from Table 20-3, 
and e p aT p 4 becomes the planet emittance W p . The instantaneous planet heat flux is: 


H' p = W p Fr 

















820 


SPACECRAFT THERMAL DESIGN 



TEMPERATURE (°R) TEMPERATURE (°R) 


Fig. 20-42. Space radiation to an insulated 
thin plate, 200°-900°R. 


Fig. 20-43. Space radiation to an insulated 
thin plate, 800°-1500°R. 



PLATE ORIENTATION ANGLE, y (deg) 

Fig. 20-44. Geometric factor for earth radiation to a flat plate, 
versus plate orientation and distance above earth surface. 
y is the plate orientation angle, the true angle between the 
plate normal and the earth-plate line, h = plate distance 
above the earth’s surface, in nautical miles. Fr = geometric 
factor for diffuse radiation from the earth to the plate. The 
flat plate irradiance is given by: H p — F H W P Btu (hr ft 2 ) -1 . For 
an average yearly albedo of 36%, H p — 70.9 Btu (hr ft 2 ) -1 . 













SPACECRAFT THERMAL BALANCE 


821 



Fig. 20-45. Geometric factors for earth albedo to a flat plate, versus plate orientation and 
distance above earth’s surface. F a y=o is the geometric factor from the earth to a plate 
parallel to the earth surface. F n y/F a y =0 is the plate orientation correction factor as a function 
of altitude, y is the true angle between plate normal and earth-plate line. 6 S is the true 
angle between earth-plate and earth-sun lines, h is the plate distance above earth’s surface, 
in nautical miles. The flat plate irradiance is given by: H — Fy =0 ( Fy/Fy= 0 )H„ . For the 
average yearly earth albedo of 36%, H a = 159.3 Btu (hr ft 2 ) -1 . 


20.7.2.4. Total Incident Space Radiation. If one sums the heat fluxes of solar, 
albedo, and planet radiation, the instantaneous heat flux to a satellite’s surface at a 
point is calculated from: 

H' = S cos 6 -I- aSF a + W p Fr 

as a function of </><■, 0 , d s , and y. 

The orbital average heat flux to a surface is calculated from: 

n=k n=k 

H = Scos$ + asS' Fa]r - + W P V hbh = H, + H a + H p 

Z-^ n Z—i n 

n=l n= 1 

Once the irradiances from space are known, along with the corresponding spacecraft 
surface thermal properties of a s , a a , and € s f c , the surface temperature (T s f c ) of a perfectly 
insulated material may be computed from: 

e s f c orT s f c 4 = H s a s + H„a (l + H p a p 

Generally, the absorption of the materials for albedo radiation is the same as the 
absorption for the solar spectrum, and for the case where the planet temperature and 
skin temperature are similar, the surface absorption for planet radiation (a p ) is the same 
as the surface emissivity; i.e., a p =e s f c . The above equation then becomes 

crTsfc 4 = ( H s + H a ) + H p 

20.7.3. Thermal Design. In general, the thermal coating is the best possible for 
the maintenance of life, electronic equipment, and structural integrity within specified 
temperature requirements. 














822 


SPACECRAFT THERMAL DESIGN 


Refer to Chapters 8 and 12 for the conditions under which materials and detectors 
operate. Other components have the following temperature ranges: 


Component 

Batteries 

Computers 

Electronics 

Gyros 

Control gas storage 


T emperature 

(°F) 

40-100 

25-100 

0-130 

60-160 

0-130 


To satisfy these requirements, the spacecraft thermal coupling to space must be 
designed to carry the internal power load and yet satisfy the limitations of temperature. 

20.7.3.1. Passive Control. These conditions can be satisfied by a passive system if 
the orientations and equipment power dissipations are known so that an a s /e t h can be 
chosen for operation within the temperature requirements. 

Figure 20-46 has been drawn to show the variation in equipment temperature as a 
function of its power dissipation and of thermal coating <x s and e t h. The figure incor¬ 
porates solar, earth albedo, and earth radiation to a flat plate for a 500-mi circular 



TEMPERATURE (°R) 


Fig. 20-46. Temperature of equipment heat sink versus equipment power dissipation P as a func¬ 
tion of heat-sink thermal coating a s /e ltl for a flat plate. The four right-hand curves are for heat 
sink normal to sun, 65% time in sun, H s = 286, H a = 1.3, H p = 21.3 Btu (hr ft*) -1 . The left-hand 
curves are for equipment heat sink parallel to the plane of the ecliptic with the plate facing north, 
H m = 0, H a = 10.53, H p = 15.39 Btu (hr ft 2 ) -1 . 500-mile earth orbit (circular) in plane of ecliptic. 




REFERENCES 


823 


earth orbit. Five sides of the equipment box are assumed to be insulated; the front 
surface (heat sink), which faces the space environment, is not. The maximum en¬ 
vironmental heat input assumes a heat sink normal to the sun, and the minimum heat 
input assumes the surface parallel to the sun’s rays. 

If one knows the power dissipation per square foot of heat-sink area, the thermal- 
coating properties of a s and e,i, can be selected, so that temperature requirements are 
not exceeded, with the unit in or out of the sunshine. 

The heat balance is calculated from the following equation, where the subscript "eq” 
means "an equipment parameter”: 

( H, + H„) ^ + H„ + ^ = aT 4 

€ih €th 


20.7.3.2. Active Control. When conditions cause the temperature to fall outside 
the required range, an active system may be employed. 

Mechanical means such as pinwheels or louvres may be used to vary the a s /e,h of the 
heat sink. Other methods such as electrical-thermostat heat controls or variable con¬ 
ductance systems alter the internal heat flow to the heat-sink surface and in like manner 
create a stabilizing effect on component temperatures. 

20.8. Testing 

Testing of spacecraft can involve either cold-wall vacuum operation, cold-wall vacuum 
plus solar simulation, or cold-wall vacuum plus solar simulation and simulation of 
albedo and earth radiation. 

A typical test facility designed to simulate true space conditions of solar, albedo, and 
earth radiation, plus vacuum and cold space is in use by the National Aeronautical 
Space Administration at Langley Field, Virginia [28]. 

References 

1. A. G. Worthing and D. Halliday, Heat, Wiley, New York, p. 435 (1948). 

2. W. N. Harrison and J. C. Richmond, Standardization of Thermal Emittance Measurements, 
WADC 59-510, Aeronautical Systems Division, Dayton, Ohio (1960). 

3. J. Ashbrook, G. F. Schilling, and T. E. Sterne, Glossary of Astronomical Terms for the Descrip¬ 
tion of Satellite Orbits. 

4. E. Fried, Thermal Joint Conductance in a Vacuum, ASME, 63-AHGT-18, Aviation and Space, 
Hydraulic, and Gas Turbine Conference and Products Show, Los Angeles (March 3-7, 1963). 

5. E. Fried and F. Costello, ARS Journal 32, 237 (1962). 

6. D. O. Kern, Process Heat Transfer, McGraw-Hill, New York, pp. 80-82 (1950). 

7. W. R. Morgan and D. C. Hamilton, Radiant-Interchange Configuration Factors, NACA TN 2836, 
Purdue University, LaFayette, Ind. (1952). 

8. M. Jakob, Heat Transfer, Vol. I, Wiley, New York (1949). 

9. W. H. McAdams, Heat Transmission, 3d ed, McGraw-Hill, New York (1954). 

10. F. S. Johnson, J. Meteorol., 11, pp. 431-439 (1954). 

11. L. D. Jaffe and J. B. Rittenhouse, ARS Journal, 32, p. 336 (March 1962). 

12. K. A. Ehricke, Space Flight; Principles of Guided Missile Design, \ an Nostrand, Princeton, 
N.J., pp. 118-119 (1960). 

13. J. C. Ballinger, J. C. Elizalde, and E. H. Christensen, "Thermal Environment of Interplanetary 
Space,” 344B, SAE National Aeronautic Meeting, New York (1961). 

14. C. G. Claus and J. B. Singletary (eds.), Space Materials Handbook, Lockheed Missiles and 
Space Co., Sunnyvale, Calif., AF 04(647)-673, pp. 32, 47, 48, 128, 129, 183 (1962). 

15. F. Sartwell, National Geographic, 123, p. 733 (1963). 

16. W. Nordberg et al., Preliminary Results of Radiation Measurements from the TIROS III Me¬ 
teorological Satellite, TND-1338, National Aeronautics and Space Administration, Washington, 

D.C. (1962). . 

17. S. B. Nicholson and E. Pettit, "Lunar Radiation and Temperature,” p. 392, Carnegie Institution, 

Washington, D.C. 



824 SPACECRAFT THERMAL DESIGN 

18. W. B. Fussell, J. J. Triolo, and J. H. Henninger, A Dynamic Thermal Vacuum Technique for 
Measuring the Solar Absorptance and Thermal Emittance of Spacecraft Coatings, TND-1716, 
National Aeronautics and Space Administration, Washington, D.C. (1963). 

19. D. K. Edwards et al., J. Opt. Soc. Am., 51, pp. 1279-1288 (1961). 

20. R. V. Dunkle et al., Heated Cavity Reflectometer for Angular Reflectance Measurements, Aca¬ 
demic Press, New York, pp. 541-562 (1962). 

21. D. K. Edwards et al., Basic Studies on the Use and Control of Solar Energy, Annual Report 
NSF-G-9505, Report 60-93, University of California, Los Angeles (Oct. 1960). 

22. R. E. Gaumer, "Determination of the Effects of Satellite Environment on the Thermal Radia¬ 
tion Characteristics of Surfaces,” Paper 339C, SAE National Aeronautic Meeting, New York. 

23. American Institute of Physics Handbook, McGraw-Hill, New York, pp. 68-71 (1957). 

24. R. E. Gaumer and L. A. McKellar, Thermal Radiative Control Surfaces for Spacecraft, LMSD- 
704014, Lockheed Missiles and Space Division, Sunnyvale, Calif. (March 1961). 

25. J. H. Weaver, Anodized Aluminum Coatings for Temperature Control of Space Vehicles, 
Technical Memorandum ASCN 62-9, Aeronautical Systems Division, Dayton, Ohio (March 
1962). 

26. W. D. Wood and C. F. Lucks, Thermal Radiative Properties of Selected Materials, DMIC Report 
177, Vol. I, Defense Metals Information Center, Battelle Memorial Institute, Columbus, Ohio 
(Nov. 15, 1962). 

27. Grumman Aircraft Engineering Corporation data. 

28. L. G. Clark and K. A. Laband, "Orbital Station Temperature Control,” Astronautics, 7, pp. 
40-41, (Sept. 1962). 

Bibliography 

Anderson, D. L., and G. J. Nothwang, Effects of Sputtering with Hydrogen Ions on Total Hemi¬ 
spherical Emittance of Several Metallic Surfaces, NASA TND-1646, National Aeronautics and 
Space Administration, Washington (January 1963). 

Clauss, F. J., (ed.), First Symposium on Surface Effects on Spacecraft Materials, John Wiley & Sons, 
New York, 1960. 

F. G. Cunningham, Power Input to a Small Flat Plate from a Diffusely Radiating Sphere with Applica¬ 
tion to Earth Satellites: The Spinning Plate, NASA TND-1545, National Aeronautics and Space 
Administration, Washington (February 1963). 

Hastings, E. C., Jr., R. E. Turner, and K. C. Speegle, Thermal Design of Explorer XIII Micrometeoroid 
Satellite, NASA TND-1001, National Aeronautics and Space Administration, Washington (May 
1962). 

Hayes, R. J., Quarterly Progress Report Determination of the Emissivity of Materials, PNA-2163, 
Pratt and Whitney Aircraft Division of United Aircraft Corp., Hartford, Conn. 

Hembach, R. J., L. H. Hemmerdinger, and A. J. Katz, Heated Cavity Reflectometer Modifications, 
ADR. 04-03-62.2, Grumman Aircraft Engineering Corp., Bethpage, L.I., N.Y. (July 1962). 

Hemmerdinger, L. H., General Description and Operation of Grumman Spectrophotometers, PM-27, 
Grumman Aircraft Engineering Corp., Bethpage, L.I., N.Y. (June 1961). 

Hemmerdinger, L. H., "Thermal Design of the OAO,” J. Spacecraft Rockets, 1,5, (Sept.-Oct. 1964). 

Henderson, R. E., and D. L. Dresser, "Solar Thermionic Space Power Systems,” Paper 350C, SAE 
National Aeronautic Meeting, New York (1961). 

Katz, A. J., Determination of Thermal Radiation Incident upon The Surfaces of an Earth Satellite in 
an Elliptical Orbit, XP12-20, Grumman Aircraft Engineering Corp., Bethpage, L.I., N.Y. (May 
1960). 

Sadler, R., and L. H. Hemmerdinger, The Emissometer — A Device for Measuring Total Hemispherical 
Emittance, PM-001-33, Grumman Aircraft Engineering Corp., Bethpage, L.I., N.Y. (Jan. 1963). 

Samela, D. A., "Thermal Analysis of the Echo II Canister Assembly,” ASME, 63-AHGT-54, Aviation 
and Space, Hydraulic, and Gas Turbine Conference and Products Show, Los Angeles, Calif. 
(March 3-7, 1963). 

Weaver, J. H., Anodized Aluminum Coatings for Temperature Control of Space Vehicles, ASD-TDR- 
62-918, Aeronautical Systems Division, Dayton, Ohio (Feb. 1963). 

Radiation Heat Transfer Analysis of Space Vehicles, ASD-TR-61-119, Part II, Aeronautical Systems 
Division, Dayton, Ohio (Sept. 1962). 


Chapter 21 

AERODYNAMIC INFLUENCES 

ON INFRARED 
SYSTEM DESIGN 

Lawrence D. Lorah and Eugene Rubin 

Mithras, Incorporated 


CONTENTS 


21.1. Introduction.826 

21.2. Infrared Windows.826 

21.2.1. Window Requirements.826 

21.2.2. Window Heating.827 

21.2.3. Typical Results.832 

21.2.4. Window Transmission and Radiation at Elevated Temperatures . 833 

21.2.5. Method of Alleviating Hot-Window Problems.836 

21.3. Refraction by the Field of Flow.839 

21.3.1. Index of Refraction of Air.839 

21.3.2. Shock-Wave Effects.839 

21.3.3. Boundary-Layer Effects.846 

21.3.4. Fluctuating-Boundary-Layer Effects.848 

21.4. Radiation from Heated Air.851 


825 
















21. Aerodynamic Influences on Infrared 

System Design 


21.1. Introduction 

This chapter discusses the following major problems encountered by supersonic 
missiles and aircraft: first, the influence of high-speed flight on the window covering 
the infrared equipment; second, the optical influence exerted by the aerodynamic flow 
field around the vehicle; and third, the background radiation produced by the heated 
air surrounding the vehicle. 

The discussions in Sec. 21.2 and 21.3 are restricted to supersonic flight over the Mach 
number range of about 1.5 to 7.0. At speeds below this, the problems considered here 
do not exist and system design requires only minor structural differences from those 
in surface-based equipment. At speeds above Mach 7 in atmospheric flight the assump¬ 
tion of perfect gas flow begins to break down and the chemistry of the air must be 
considered. The nature of the aerodynamic effects on infrared systems is the same at 
these higher speeds, but the details of predicting these effects are much more complex 
and beyond the scope of this discussion. In this chapter the ratio of specific heats for 
air is taken as 1.4 and the numerical coefficients are evaluated accordingly. In Sec. 
21.4 the chemistry of the air is a factor and the limitation to Mach numbers below 7 
no longer exists. 

21.2. Infrared Windows 

Infrared windows are basically protective coverings over the cavities or bays that 
house infrared or optical equipment. Generally, they are not an integral part of the 
functional scheme of the equipment and any change they make in radiation falling on 
the system produces a degradation of system performance. Satisfactory windows for 
ground-based equipment or low-speed aerial equipment can generally be designed by 
careful selection of a window material from those of Chapter 8 and then applying com¬ 
mon optical and structural design techniques. 

21.2.1. Window Requirements. The primary purpose of the window, whether it 
be a missile nose dome or a flat reconnaissance window well back along the aircraft 
fuselage, is to protect infrared equipment from the aerodynamic loads and severe 
heating rates encountered in supersonic flight. While performing this basic function, 
the window must meet the following additional requirements: 

1. The window must introduce negligible distortion in the image formed by the 
optical system. This consideration nearly always results in a window which 
is a portion of a sphere or is made with one or more flat surfaces. Other shapes 
(cones, ogives, etc.) give large refraction errors and resolution losses unless the 
window is made very thin (see Chapter 8). However, windows that are too thin 
cannot carry the necessary structural loads. 

2. The window must transmit radiation in the appropriate wavelengths without 
gross attenuation. The usual window material selection is complicated in this 
case by the fact that many materials lose transparency as their temperature 
increases, particularly near the long-wavelength cutoff (see Chapter 8). 


826 


INFRARED WINDOWS 


827 


3. The window must radiate a negligible amount of energy in the wavelength region 
of system operation. Even though the window emissivity is low in these wave¬ 
lengths (by virtue of requirement 2) the window proximity to the sensing system 
and the high window temperatures achievable in supersonic flight make this a 
prime consideration. Window radiation can saturate the detector, and non- 
uniform radiation from the window can produce false signals which will mislead a 
tracking system, or produce ghosts in a reconnaissance system. 

4. The window must not impose an unreasonable drag penalty on the vehicle. 

5. The window must withstand the mechanical and thermal loads induced by high¬ 
speed flight. 

Other factors such as ease of fabrication, light weight, ground handling loads must 
also be considered. Most of these requirements depend in whole or in part on the 
aerodynamic heating experienced in supersonic flight. 

21.2.2. Window Heating 

21.2.2.1. Aerodynamic Heat Transfer. The rate of aerodynamic heat transfer per 
unit area, q , of the wall (window) w can be written 

q = h(T r -T w ) (21-1) 


where h = heat-transfer coefficient 
T w = wall temperature 
7V = recovery temperature 

The heat-transfer coefficient is a function of the wall geometry, the flight conditions, 
and to some extent the wall temperature. The recovery temperature is equal to the 
temperature of the wall when there is no aerodynamic heat transfer. This temperature 
is usually defined by means of a recovery factor r: 

7V= 7X1 + 0.2rM 2 ) (21-2) 

If the wall (window) is assumed to be thermodynamically thin, i.e., 


and the radiation and conduction losses are negligible, the aerodynamic heat-transfer 
rate can be equated to the rate of enthalpy increase within the wall, or 

dT w 

h(T r — T w ) = p w b w c p ~ 

w dt 


In the last two expressions, b = window thickness, k — thermal conductivity, p density, 
and t = time. 

If there is a step change in the aerodynamic conditions, e.g., velocity (thus a step 
change in h and T r ), the solution is 


T r -T 


Tr 


-Ti 


= exp — 



(21-3) 


where T, = initial temperature at t = 0 






828 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


At the higher Mach numbers (higher window temperatures) the radiation heat trans¬ 
fer plays a significant part. If this is to be considered, the heat-balance equation 
becomes 

h(T r — T w ) = Pvbuc, d ^f+ <n,{T)T«* ( 21 - 4 ) 

w at 

where the equivalent total wall emissivity e P is a strong function of temperature because 
of its large spectral variation and a = Stefan-Boltzmann constant. Equation (21-4) 
can be integrated step by step to yield a more accurate temperature history. The true 
equilibrium temperature ( dT/dt = 0) can also be found from Eq. (21-4). 

In a range of aerodynamic variables, it is more convenient to normalize the heat- 
transfer coefficient with respect to the local free-stream properties. The normalized 
heat-transfer coefficient is the Stanton number (St), 

St = —-— (21-5) 

PelleCp 


where u = velocity and e = flow properties of external edge of boundary layer. 

For a more rigorous treatment of aerodynamic heat transfer, see [ 1] and [2]. Details 
of the temperature distribution within windows, given the aerodynamically imposed 
boundary conditions, can be developed by methods described in [3] and [4]. 

21.2.2.2. Heat Transfer to Flat Plate. The boundary-layer flow over a flat surface 
can be either laminar or turbulent. In most flight situations where the flat window is 
more than a foot or so from the nose or leading edge of the vehicle, the flow is turbulent. 
Transition from laminar to turbulent flow takes place at Reynolds numbers Re of ap¬ 
proximately 10 5 to 5 X 10 6 , the lower value being more likely in an actual flight vehicle 
with the usual surface roughness. 

In the laminar case, the results of Van Driest [5] can be used directly. These exten¬ 
sive heat-transfer calculations are summarized in Fig. 21-1 as curves of St versus 
Mach number for various ratios of wall to free-stream temperature. 


T /T 

W' 00 



Fig. 21-1. Local heat-transfer coefficient for 
laminar boundary layer of a compressible fluid 
flowing along a flat plate, T m = 400°R [5], 














INFRARED WINDOWS 


829 


Experimentally, the recovery factor, r, is given for laminar flow as VFr, or about 
0.848 as an average over normal flight conditions ( Pr = Prandtl number). 

In the turbulent case, use is made of the Reynolds analogy, which relates the Stanton 
number to the local skin friction coefficient c f through the equation 

s '~~2 ( 21 - 6 ) 

Here 1/s is a measure of the accuracy of the analogy. For current purposes assume 
5 — 1 alt'hough in some cases s can be as low as 0.8. Extensive measurements of skin 
friction have been made, and a semi-empirical relation which holds generally for in¬ 
compressible turbulent flow has been found [6] which adequately predicts the local cy: 

Cfi = 0.370/(logio Rej-) 2 584 (21-7) 


where i = incompressible and x = some arbitrary distance. 

Flow compressibility has a marked influence on the friction coefficient. To account 
for this, the ratio of compressible to incompressible friction coefficient for air is [1,6] 


c h 




+ 0.0394M 2 


- 0.648 


( 21 - 8 ) 


In the special case where the wall temperature is equal to the recovery temperature 
(insulated wall case) 


c h 


1 + 0.129A/ 2 


- 0.648 


(21-9) 


The turbulent recovery factor, r, is approximately (Pr) 113 , or about 0.896. 

The insulated wall case serves as a handy reference for heat-transfer calculations. 
At T w = T r there is, of course, no heat transfer, however, the Stanton number for the 
insulated case can be computed for these conditions. This corresponds rigorously to 

St (pelleCp )- 1 


at T w = T t and can be used to determine the heat-transfer coefficients at wall tempera¬ 
tures in the vicinity of 7V. Figure 21-2 shows the variation of both laminar and turbu¬ 
lent Stanton numbers as a function of Mach and Reynolds numbers. 

21.2.2.3. Heat Transfer to a Hemisphere. The aerodynamic flow to a hemispherical 
nose at the stagnation point (i.e., the foremost point on the hemisphere, regardless of 
angle of attack), is laminar. As the air accelerates in passing from the stagnation 
point to the base of the hemisphere, it can go through transition and the boundary 
layer generally becomes turbulent. This happens at Reynolds numbers around 4 x 10 5 . 

For the laminar case, Sibulkin [7] has shown that locally the Stanton number on a 
hemisphere stagnation point can be written 


St = 0.763 





( 21 - 10 ) 








830 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 



Fig. 21-2. Stanton number as a function of Reynolds number for an insulated plate. 


where D = diameter, /3 = velocity gradient, p = viscosity, and all of the fluid properties 
are evaluated at the local conditions just outside the boundary layer. In the case of a 
stagnation point ( s.p .) this corresponds to stagnation temperature and pitot pressure. 
This reduces Eq. (21-10) into terms of free-stream flow: 


St = 0.763 




( 21 - 11 ) 


where (c p plk) s . p . = stagnation Prandtl number (approximately 0.72 for normal flight 

conditions) 

pJJDIp x = free-stream Reynolds number based on hemisphere diameter 
From [1], 



If it is assumed that p T°- 7e and the perfect gas equations of [8] are used, Eq. (21-11) 
then reduces to the general laminar Stanton number equation 


St VRep VU/D = 9.37 


M 2 


M 2 + 5 


0.25 


1 + M ^°- 38 


( 21 - 12 ) 


This gives the heat-transfer rate at the stagnation point where the recovery factor, 
r, equals 1. The laminar heat-transfer rate decreases away from the stagnation point. 
Reference [9] gives methods for determining the laminar heat-transfer distribution 
over the surface of the hemisphere. 










INFRARED WINDOWS 


831 


When the boundary layer is turbulent, the Stanton number has a maximum at about 
40° from the stagnation point, then decreases further aft. To compute the local turbu¬ 
lent heat-transfer rate on the face of the hemisphere the "local flat plate” method can 
be used with good accuracy. 

This approach calls for utilization of Eq. (21-6), (21-7), and (21-8) to determine the 
local Stanton number. All of the flow properties are evaluated at "local” conditions, 
that is, at the pressure, velocity and temperature just outside the boundary layer at any 
angular station 0. The Mach number distribution over the face of a hemisphere is 
insensitive to free-stream Mach number (Fig. 21-3). The local pressure is given by [8]. 



(21-13) 


where 


P 02 = (6Mx 2 \ 712 / 6 \ 5/2 

P v ~ V 5 / \1M\ - 1/ 



Fig. 21-3. Local Mach number on hemisphere. 


The local static temperature 



oo 


(21-14) 


since the stagnation temperature is unchanged by a shock wave. 










832 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


Re 

X 

r d 

(T/T 0 )e - 0 

o2.74X 10 6 

19.26 x 10 6 

0.49 

•3.43 

18.06 

.58 

•3.45 

18.01 

.58 

•3.50 

10.85 

.77 

•4.38 

13.04 

.79 

•4.45 

17.09 

.69 


-Laminar theory 

-Local flat plate 

(turbulent theory) 



50 100 
& (degrees) 


Fig. 21-4. Variation with d of the ratio of local heat-transfer 
coefficient to the value at the stagnation point at M x = 2.0 and 
several free-stream Reynolds numbers [10]. 


A typical case is shown in Fig. 21-4, where both theoretical and experimental heat- 
transfer distributions are shown [10]. 

21.2.3. Typical Results. The following subsections apply some of the basic equa¬ 
tions previously discussed to typical flight situations to determine what sorts of temper¬ 
atures and heating rates can be expected. 

21.2.3.1. Maximum or Recovery Temperature. If there are no conduction losses and 
the window emissivity is zero, the recovery temperature can be determined directly 
from Eq. (21-2) by supplying the appropriate recovery factor, 0.848, for laminar flow 
and 0.896 for turbulent flow. However, if the radiation heat transfer cannot be ne¬ 
glected, Eq. (21-4) must be solved, with dTJdt = 0 at equilibrium. 

The actual total emissivity of the window depends strongly on the particular material 
selected, and on the instantaneous temperature. However, here e w will be considered 
constant. A total emissivity of 0.25, over all wavelengths, is characteristic of some 
materials over a large temperature range. Figure 21-5 shows the maximum or equilib¬ 
rium temperatures at three altitudes and at Mach numbers up to 7 for the flat plate, and 
for a hemispherical window 6 in. in diameter. 

21.2.3.2. Rate of Temperature Increase. To give an indication of the relative rate of 
temperature increase, Eq. (21-3) is used. The equation for a time constant, r, is 


and gives the time to reach 



T r -T 1 
T _ T - e ~ 0.368 


(21-15) 


(21-16) 











INFRARED WINDOWS 


833 



Fig. 21-5. Maximum temperature due to aerodynamic heating, 
turbulent flat plate, x = 1 ft. 


o 


S 


o 


eg 


bD 

s 


o 


o 

0 ) 

(fl 

i 


100 


10 



MACH NUMBER 


Fig. 21-6. Time constant for aerodynamic heating. 


in the absence of radiation. In Fig. 21-6 the generalized time constant is given for the 
conditions of Fig. 21-5, and a typical material, MgF 2 , 0.07 inch thick, is used to get a 
specific time constant. 

21.2.4. Window Transmission and Radiation at Elevated Temperatures. Aero¬ 
dynamic heating can produce changes in the infrared characteristics of window mate¬ 
rials. A major problem at elevated temperatures is a loss in transparency, particularly 
at the longer wavelengths. Transmission characteristics of materials at high tempera¬ 
tures are covered in Chapter 8. Any body, semi-transparent or opaque, will radiate 
thermal energy in ever-increasing amounts as the temperature rises. Two different 
types of effects are produced by a radiating window: 













834 


AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


(1) Because of strong temperature gradients, or local "hot spots,” false targets can 
appear. 

(2) A large irradiance on the infrared detector, even though uniformly distributed, 
can saturate the detector and decrease its sensitivity. 

The susceptibility of a particular guidance system to false targets associated with 
nonuniform temperature distributions is greatly dependent on the details of the dis¬ 
crimination and tracking methods employed. A window which is nearly hemispherical 
does not generally have large temperature gradients along its surface, even though 
the temperature level can be quite high. A pointed window, such as a cone, can have 
very high temperature gradients along its surface, particularly when used at high 
speeds and moderate altitudes. Any small protuberances optically distort the in¬ 
coming radiation and produce local hot areas on the window. Generally, these de¬ 
viations are spread out by the thermal conductivity of the window and are close enough 
to the optical system to be out of focus. A point on the window is not likely to appear 
as an exact replica of a target, but will produce additional noise in the tracking system, 
thereby effectively reducing its sensitivity. 

The problem of saturation of the infrared detector by the window radiation is usually 
more critical than the "false target problem” and can be treated more precisely. Details 
of the particular optical techniques employed do not influence the saturation calculation, 
and a general analysis is possible. If the field of view is assumed to be small, and the 
window is assumed to be isothermal and fills the entire field of view, the radiance B 
can be expressed as 


B = ~ f Wk t» edK (21-17) 

Now the effective radiant power P incident on the collector is 

P = BA c 7r(n) 2 (21-18) 


where O is the half-angle of the field of view. By neglecting the losses in radiant power 
due to any space filtering, refractive optical elements, etc., the effective radiant power 
falling on the detector is equal to that falling on the collector. Therefore, the irradi- 
ance-at the detector is 


P_ 

Ad 


Hd — — = Bird 2 — 


Ac 

Ad 


(21-19) 


By geometry 


Ac 

Arf 


;(?)' 


n 2 


where /= focal length. The irradiance at the detector is a function of only the window 
radiation and the focal length-to-diameter ratio of the collector: 



( 21 - 20 ) 


INFRARED WINDOWS 


835 




ee 


>7 

H 

H-< 

> 

*—* 

C/3 

C/3 

►—« 

s 

w 



(c) 


30 


0 


0 1.0 



2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 

WAVELENGTH (p) 


Fig. 21-7. Emissivity of (a) germanium (single crystal, 1.14 mm thick); 
(6) silicon (single crystal, 4.16 mm thick); (c) synthetic sapphire (4.11 mm 
thick) [11]. 


The emissivities of several window materials are shown in Fig. 21-7, others are 
covered in Chapter 8. 

Figure 21-8 shows the detector effective spectral irradiance due to a typical window of 
MgF 2 at various temperatures. 

In a conventional air-to-air missile application, the window temperature can reach 
several hundred degrees. Figure 21-8 shows that at the detector the background 
irradiance caused by the window is many orders of magnitude greater than the ir¬ 
radiance from the target. The saturation effects of elevated window temperature are 
shown in Fig. 21-9 as the relative magnitude of the minimum detectable signal (MDS), 
normalized for MDS at a window temperature of 50° C, as a function of window tempera¬ 
ture. It is assumed that no window radiation outside the 3-/x to 5-/x region falls on the 















836 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 



Fig. 21-8. Effective spectral irradiance at a 
detector with an MgF 2 window vs. wave¬ 
length, at various temperatures. 



WINDOW TEMPERATURE (°C) 


Fig. 21-9. Relative minimum detectable 
signal for Ge:Au and PbTe detectors as a 
function of MgF 2 window temperature. 


detector. A PbTe detector can lose sensitivity by a factor of 8 if the MgF 2 window 
temperature is raised from 50° C to 700° C, and the Au:Ge detector can lose sensitivity by 
a factor of 360 for the same window temperature change. It should be noted that MgF 2 
is a good infrared window material between 3 p. and 5 p, and other materials would 
probably produce more of a saturation effect over the same temperature range. 

21.2.5. Methods of Alleviating Hot-Window Problems 

21.2.5.1. Optical-System Parameters. Several optical-system parameters have a 
considerable influence on the overall performance through the effects of window radia¬ 
tion. They include: 

(1) Spectral region of sensitivity — the spectral distribution of the window radiation 
can alter the optimum wavelength limits from those indicated by target-detector- 
background considerations alone. 

(2) Detector type —of the several detectors that can operate in the desired spectral 
region, improved saturation characteristics can produce a greater signal-to-noise 
ratio than slightly greater detectivity in the presence of strong window radiation. 

(3) Field of view — reduction of the field of view is important since the window 
irradiation on the cell and the cell area are reduced. 

(4) Window location —for systems not requiring a nose installation, careful selection 
of the window location so as to avoid regions of high heat transfer (shock-wave 
impingement, boundary-layer reattachment, etc.) can reduce window radiation. 







INFRARED WINDOWS 


837 


21.2.5.2. Delay in Temperature Rise of the Window. In many systems, the time of 
operation is short enough and/or the closing rate on the target is rapid enough (the 
target signal increases) so that a simple delay in the temperature rise of the window 
constitutes a satisfactory solution to hot-window problems. 

Several delaying methods can be used. The most obvious approach is to precool the 
window. When the tactical situation permits {i.e., internal missile storage in flight, 
ground launch) a marginal condition can be made operational without complicating the 
basic system design. 

Assuming that the window material has the highest value of pc p consistent with the 
optical requirements, the time constant can be lengthened by increasing the window 
thickness, 6. This is fairly effective with window materials in which the transmission 
loss is due primarily to reflection rather than absorption. Optical distortion and 
window weight also limit this approach. Again, a marginal condition can be improved 
by this method. 

Another solution involves reducing the heat-transfer coefficient. A flat window 
mounted along the side of a vehicle can be recessed below the vehicle surface, thereby 
separating the boundary layer from the window surface. 

Assuming that the cavity depth is of the same scale as the thickness of the boundary 
layer and that the boundary layer is always turbulent, two separate flow configurations 
can exist over such a shallow cavity. The first, "closed cavity,” flow exists when the 
boundary layer, initially separated as it crosses the upstream edge of the cavity, attaches 
to the cavity floor and then separates again ahead of the downstream edge of the cavity 
(Fig. 21-10a). The second, "open cavity,” flow exists when the boundary layer remains 
separated over the entire length of the cavity (Fig. 21-106). The flow configuration goes 
from closed to open as the cavity length-to-depth ratio is decreased. As the boundary 
layer becomes thicker with respect to the cavity depth, the transition from closed flow 
to open flow becomes more gradual and the flow configuration is not as sensitive to 
changes in length-to-depth ratio, LIH. Roughly, for a uniform depth cavity, length-to- 
depth ratios less than about 10 will produce open-cavity flow. 



pi ow Turbulent Boundary Layer 





(b) 


Fig. 21-10. Flow configurations, (a) Closed cavity; ( b ) open cavity. 

























838 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


Theoretical treatment of aerodynamic heat transfer in separated flow has not proved 
too satisfactory because of lack of available experimental data. For experimental data 
measured in a configuration similar to the one under consideration at moderate Mach 
numbers, see [12]. The heat-transfer coefficient, h, on the cavity floor, normalized 
with respect to the heat-transfer coefficient for an attached flow at the same body station 
and Reynolds number, is shown in Fig. 21-11 and 21-12. 




X/L X/L 


(a) (b) 

Fig. 21-11. Variation of the heat-transfer distribution on the floor of the 
cutout as a function of LIH at M 0 = 2.9 [12]. (a) Thin oncoming boundary 
layer; (b) thick oncoming boundary layer. 



(a) (b) 

Fig. 21-12. Parametric map of heat-transfer coefficient ratio (minimum, 
maximum, and at XIL = 0.7) as functions of LIH and M 0 , (a) thin boundary layer; 
( b ) thick boundary layer [12], Note: points are averages over all tests. 











REFRACTION BY THE FIELD OF FLOW 


839 


For all values of L/H the heat-transfer-coefficient ratio drops considerably below 1 
in the separation wake, and rises above 1 in the recompression wake, reaching a maxi¬ 
mum at the downstream edge of the cavity. The typical h/h f distributions plotted in 
Fig. 21-11 show that an increase in the boundary-layer thickness does not change the 
general character of the curves, but does raise significantly the average level of heat 
transfer. In Fig. 21-12 it can be seen that the relative heat-transfer coefficient de¬ 
creases with decreasing Mach number and a higher ratio of LIH produces a greater 
maximum heat-transfer coefficient. 

21.3. Refraction by the Field of Flow 

21.3.1. Index of Refraction of Air. In the analysis of aerodynamic influences on 
electromagnetic radiation, the index of refraction n, of air must be considered. The 
major change in index of refraction of air takes place with a change in density, p. 
The variation of index of refraction for transparent substances as a function of density 
is [13] 

iri= kp (21 - 21) 

This is the Lorenz-Lorenz law which for gases (n ~ 1) reduces to the simpler Gladstone- 
Dale law: 

n - 1 = kp (21-22) 

with virtually no loss in accuracy. Figure 21-13 shows the variation of k in Eq. (21-22) 
as a function of wavelength, as deduced from the data of [14]. Variations in index of 
refraction due to temperature effects on molecular structure and changes in atmospheric 
composition are neglected. 

Figure 21-13 shows that the effects of refraction through nonuniformities in air are 
more pronounced in the visible portion of the spectrum than in the infrared. In the 
wavelength region beyond the near infrared (X > 1 p) the refractive index of air is 
nearly invariant with wavelength; thus one finds no dispersion and no chromatic 
aberration. 



Fig. 21-13. Gladstone-Dale constant for air [14], 

21.3.2. Shock-Wave Effects. As a body travels through the atmosphere, the air 
must adjust to accommodate it. At supersonic velocities the initial adjustment takes 
place across shock waves, which are discontinuities in the air properties.* Surfaces 

*Shock waves are not exactly discontinuities but do have a finite thickness. This thickness is 
about five molecular mean free paths, which allows translational and rotational equilibrium of the 
molecules to be established. At sea level the shock wave is about 1 p thick. 







840 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


with sharp leading edges at moderate, positive angles of attack have these shock waves 
attached to the leading edge; relatively blunt bodies have shocks detached and standing 
some distance ahead of the body. The strength and orientation of these shock waves 
are determined by the flight Mach number and the geometry of the body. 

The relation between the density ahead of the shock, p u and the density behind the 
shock, p 2 , is 


P 2 _ 6MJ sin 2 0 
p, MJ sin 2 0 + 2 


(21-23) 


Equation 21-23 shows that the density ratio (shock strength) increases with the shock 
angle, 0, and the free-stream Mach number, M, approaching the maximum value of 6 
for a normal shock at M x . 

No convenient, explicit relations exist for determining the shock angle. However, 
curves of shock angle as a function of Mach number and flow inclination are given in 
[8]. Figures 21-14 and 21-15 show representative curves of the functions for both 
wedge and conical flow. 



Fig. 21-14. Shock-wave angle for conical flow. 



DEFLECTION ANGLE 6 (degrees) 

Fig. 21-15. Shock-wave angle for wedge flow. 








REFRACTION BY THE FIELD OF FLOW 


841 


A shock wave produced by solid surfaces is exceptionally steady and, because of the 
density change across it, is a good refracting surface. Since a plane shock wave is not 
necessarily perpendicular to the optical axis or parallel to the window surface, the 
effects of refraction must be taken into account even in the simplest case. Using Snell’s 
law, the angle (3i of the incident ray and the angle /3 2 of the refracted ray, both measured 
from the normal to the shock plane, can be related by Eq. (21-24): 


or alternatively, 


sin /3i _ 1 4- kp 2 
sin 1 4- kpi 


sin /3 i _ 1 rii — 1 p 2 
sin /3 2 rii n\ pt 


(21-24) 


(21-25) 


where n\ is a special index value in the first medium. From Eq. (21-25), the angular 
deviation of a ray passing through a plane shock wave can be calculated as a function 
in incidence angle. This is shown in Fig. 21-16 for various density ratios and two 
altitudes. Figure 21-16 shows that the density ratio of 6 (the limiting value) at sea 
level is the worst possible refraction error. The angular deviation in this limiting 
case is seen to be about 1.5 mrad (300 sec of arc) at a 45° incidence angle. At the higher 
altitude, 50,000 ft, the maximum refraction error at 45° incidence is only 0.25 mil 
(50 sec), or about one-sixth of the sea-level value. The minimum density ratio shown is 
2, which corresponds to a normal shock wave at about Mach 1.6. The deviation of the 
rays is only about one-fifth of the limiting value. 



Fig. 21-16. Angular deviation of infrared radiation 
passing through plane shock waves. 













842 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


Another way of considering the variation of plane shock-wave refraction effects is 
to fix the body geometry and allow the Mach number to change, producing variations 
in density ratio and shock angle. Selecting a flow inclination of 10°, 


cos (9 4- </>i) _ 1 + kpo 
cos (9 + <f> 2 ) 1 + fcpi 


(21-26) 


Figure 21-17 shows the angular deviation of a ray as a function of its angle with respect 
to the direction of flight. This corresponds directly to the error between the true and 
apparent target position as seen by an optical system situated in the wedge shown in 
the figure. From Fig. 21-17 it can be seen that a target along the direction of flight will 
appear 0.2 mil (42 sec) too high at Mach 2, and about 1 mil too high at Mach 5. In the 
limiting case of a Mach number of infinity, the error along the flight direction is 3.2 
mils. This error is less as the incident ray becomes more nearly perpendicular to the 
shock wave, and when it is exactly perpendicular the error goes to zero. 

Because of this shock wave, wide-angle photography suffers a distortion in the flight 
direction, even if the optical axis is perpendicular to it. If this distortion is evaluated 
as the difference in angular deviation of the rays at each edge of the field of view, the 
minimum distortion occurs when 8 + <f>i = rr/2. At that point the slopes of the curves of 
Fig. 21-17 yield a distortion of about 0.3 mil per radian field of view at Mach 2, and about 
0.7 mil per radian field of view at Mach 5. More explicitly, for example, a 1-rad field-of- 
view instrument, in a 10° wedge flying at Mach 5, would show two objects, one at each 
limit of the field of view, too close together by at least 3.6 ft for every mile of range. 



Fig. 21-17. Angular deviation of infrared radiation 
passing through plane shock waves produced by a 10° 
wedge (at sea level). 










REFRACTION BY THE FIELD OF FLOW 


843 



Fig. 21-18. Conical shock wave refraction of incident 
radiation. 


The curved shock wave occurs when the shock is detached from the leading edge or 
when the body itself presents a nonplane surface to the flow ( e.g ., curved cross section, 
or body width not large compared to its thickness). The introduction of curvature into 
the density discontinuity produces a focusing effect which, except in special cases, is 
astigmatic. To demonstrate the nature of the problem, consider first the conical shock- 
wave influence on an incident plane wave of radiation approaching perpendicular to the 
cone axis, as shown in Fig. 21-18. This curved shock acts as a lens and tends to focus 
incoming rays of parallel light. For the particular case of Fig. 21-18, the focal length 
is given by 


/ _n x tan 6 
x n 2 n i 


(21-27) 


where x = axial distance from the apex of the shock cone 
6 = the shock angle. 
n 2 — index value in second medium. 

The locus of the "focal line” has been calculated for two cone half-angles, 10° and 20°, 
and the variation with Mach number is shown in Fig. 21-19. It can be seen that the 
shock lens becomes stronger {fix less) with increasing cone angle and Mach number 
because of the increase in shock strength. The effect of this conical lens is to skew the 
focal plane of a normal optical system whose axis is aligned with the incoming radiation. 
In order to estimate the importance of this effect, consider an //1 optical system with a 
15-cm aperture, 40 cm from the apex of a 10° half-angle cone. If the source is a line 
parallel to the cone axis and at an infinite distance, the change in the image at the focal 
plane can be calculated only on the basis of geometrical changes {i.e., the change in 
convergence angle of the rays). The magnitude of the defocusing due to the shock then 
can be compared to the diffraction limit of the simple optical system. Table 21-1 shows 
that the image defocusing due to the conical shock wave at Mach 3, in an uncorrected 
system, would be about five times as large as the theoretical resolution limit, and about 
ten times larger at Mach 5. Simply refocusing the system would bring the defocusing 
to within a factor of 2 of the theoretical resolution limit even at Mach 5. As the optical 
axis tilts toward the shock apex, the curvature normal to the incoming wave front be¬ 
comes elliptical and then parabolic rather than circular, introducing further astig¬ 
matism. Also, as the optical path passes through regions near the apex of the shock 
cone, the radius of curvature of the shock decreases, producing a lens of greater power. 










844 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 



Fig. 21-19. Ratio of focal length to 
axial distance from apex for a conical 
shock wave. 


Table 21.1. Effect of Conical Shock Wave on 
fjl, 15-Cm Optical System 

Mach = 3.0 Mach = 5.0 


Theoretical resolution limit 2.4 x 10 5 rad 2.4 x 10 5 rad 


Change (in focal length) 
due to shock 

Angle subtended at 
image by 15-cm focus 
with shock 

Maximum image angle 
subtended at optimum 
focus with shock 


(fwd) 2.4 x 10 -3 cm 6.4 X 10 -3 cm 
(aft) 1.7 x 10~ 3 cm 4.6 x 10~ 3 cm 


(fwd) 1.6 x 10 -4 rad 4.3 x 10 -4 rad 
(aft) 1.1 x 10~ 4 rad 3.1 x 10 -4 rad 


2.1 x 10 -5 rad 5.8 x 10 5 rad 


A more complicated optical mechanism exists in the case of flow about a conical body. 
This arises because the density of the air is not constant in the flow field between the 
shock wave and the body surface. To accurately compute this type of effect involves a 
complicated integration process; the approximate solutions are given in [15]. Typ¬ 
ically, for a cone half-angle of 20° the contributions of the density gradients behind the 
shock wave are approximately 26% and 14% of the total error at Mach numbers of 2.0 
and 3.5, respectively, when the ray is approximately perpendicular to the cone axis. 

In the case of a hemisphere traveling at supersonic speeds, a nearly spherical detached 
shock forms ahead of the body and becomes part of the optical path. Assume, for exam¬ 
ple, a plane wave of radiation falling on the hemispherical shock along its axis of 
symmetry (Fig. 21-20). Rays striking an oblique portion of the shock surface should 
be brought to some sort of focus on the axis of symmetry. However, the change in index 
of refraction across the shock is not constant but varies with the shock angle. It 





REFRACTION BY THE FIELD OF FLOW 


845 



Fig. 21-20. Hemispherical shock wave refraction of incident radiation. 


appears that each elemental annular ring, centered on the axis of symmetry, has its 
own focal length along the axis. Using the relations for the density change across the 
shock and Snell’s law, this focal point can be computed as a function of the entrance 
angle, i//. Figure 21-21 shows the results of such calculations. For Mach 3 flight at 
sea level, the focal length increases with \\t, but at Mach 5 it decreases with increasing \}j. 
Apparently, at some Mach number for sea-level flight the hemispherical shock behaves 
as a proper lens and has a single focal length. This Mach number appears to be about 
3.26. The focal length becomes shorter as the Mach number increases because of the 
stronger shock. The simplified assumption of a spherical shock front obscures the de¬ 
tailed ray behavior due to the change of shock geometry with Mach number; however, 
this is of minor importance. 




FOCAL LENGTH j_ 

SHOCK RADHJS ’ R 

Fig. 21-21. Ratio of focal length to shock radius for 
hemispherical shock wave, A. > 15 /n, at sea level. 


An additional optical element with a focal length equal to about a thousand times 
the shock radius is still significant. Consider, the same 15-cm, //I optical system 
behind a 25-cm-diameter hemispherical shock. Again, the geometrical growth of the 
point’s image in the focal plane is compared to the theoretical limit of resolution of the 
system. Table 21-2 shows the effects of the hemispherical shock wave at Mach 3 and 
Mach 5. The geometrical blurring of the image by the shock in the uncorrected focal 
plane is seen to be 50 to 100 times greater than the circle of confusion. By moving the 
focal plane a few tenths of a millimeter to obtain an optimum focus in the presence of 

















846 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


Table 21-2. Effect of Hemispherical Shock Wave 
on // 1, 15-cm Optical System 


Theoretical resolution limit 

Angle subtended by image at 
15-cm focus with shock 

Angle subtended by image at 
optimum focus 


M = 3.0 M = 5.0 

( radians ) ( radians ) 

2.4 X 10- 5 2.4 X 10~ 5 

10~ 3 1.4 X 10- 3 

1.5 X 10- 5 0.6 X 10- 5 


the shock wave, the geometrical spreading of the image becomes less than the original 
circle of confusion. If the optical system looks off the axis of the shock system, the 
situation is far more complex. The density changes lose their axial symmetry, and the 
intercepted section of the shock wave is no longer hemispherical but more nearly 
parabolic. 

21.3.3. Boundary-Layer Effects. The boundary layer can have any of several 
effects on the radiation passing through it. Steady-state effects can be similar to those 
found in the inviscid flow field, and the fluctuating nature of a turbulent boundary layer 
can have very serious optical consequences. Since the static pressure across a boundary 
layer is nearly constant, the variations in index of refraction are due entirely to the 
variation in static temperature. In the normal situation, n decreases from its free- 
stream value as the wall is approached. A minimum index of refraction would cor¬ 
respond to recovery temperature at the wall (Eq. 21-2), and this describes the "maximum 
effect” which can be expected (unless the wall is artificially heated by some means). 
It should be noted that boundary-layer thicknesses range from 0 in. at the leading edge 
of a sharp-edged surface to several inches well back along a full-scale aircraft. The 
approximate boundary-layer thickness 8 on a flat plate is given in [2] as 


Siam = 5 Wvxfu 


8 ,urb = 0.37 


-1/5 


(21-28) 


(21-29) 


where u = velocity, v = kinematic viscosity, and x = surface distance. It is seen that 8 
is proportional to x 112 in the laminar case and 8 is proportional to x 4lb in the turbulent 
case. 

On the surface of a flat plate, two optical effects in the stream wise direction can be 
caused by this boundary-layer growth. First, the streamwise curvature of the boundary 
layer can cause a focusing effect, but some simple calculations show this to be negligible. 
At Mach 2 at sea level, for example, the focal length of the laminar boundary layer, 
only 30 cm from the leading edge, is greater than 10 10 cm. The second effect is an 
angular deviation due to the locally nonparallel surfaces of the boundary layer (i.e., 
the surface at the wall and the outer edge). Under the conditions mentioned above, 
the prismlike deviation is about 1 /otrad. Increased velocity and altitude both decrease 
the magnitude of this effect. 

The curvature of a body and the resulting curvature of the boundary layer can cause 
another type of focusing to take place. Considering only curvature and refraction in 



REFRACTION BY THE FIELD OF FLOW 847 

a single plane, the apparent distances of an image from the circularly curved interface 
between two media of differing index of refraction can be written 


n\ n2 
St S 2 



(21-30) 


where R = radius of curvature of the interface 

Si and S 2 = the apparent distances as seen from within the two media 

If the boundary layer is assumed to be thin compared to the radius of curvature, 
(i e ., the radius of curvature of the outer edge of the boundary is assumed equal to the 
inner radius), Eq. (21-30) can be used to determine the refractive effects of the boundary 
layer. When Eq. (21-30) is applied successively to thin layers of constant index of 
refraction, it is found that the indices of refraction at the wall and at the outer edge of 
the boundary layer will completely describe the refractive power of the boundary layer. 
By setting the original source distance at infinity, the incoming rays are made parallel, 
and the resulting apparent source distance from the boundary layer is the effective 
focal length. Thus, 


/ _ n w 

R ( Tt%o fix') 


(21-31) 


The index of refraction in the local free stream, n x , can easily be determined once flight 
conditions have been established; however, the index of refraction at the wall, n^ 
cannot be precisely predicted unless some knowledge of the wall temperature is at hand. 
As an example, the maximum effect will be calculated {i.e., wall temperature equals 
recovery temperature). In general, the density of the boundary layer will be less at 
the wall than at the free-stream edge, and the minimum density possible is that asso¬ 
ciated with recovery temperature for a turbulent boundary layer. Recalling that the 
static pressure is constant across the boundary layer, Eq. (21-2) and the perfect gas law 
can be used to find the density difference. With these results and the Gladstone-Dale 
law, Eq. (21-31) can be rewritten 

£ = - (5r-’M- 2 + 1) - 5r-W- 2 (21-32) 


From Eq. (21-32) it can be seen that the focal length is negative, which means that the 
boundary layer acts as a concave lens and makes incoming parallel rays diverge. In 
addition, the larger the absolute value of the ratio ///?, the less the effect that the 
boundary layer has on optical performance. At low Mach numbers the boundary layer 
is not a very important optical element, and at Mach numbers above 5 or 6 the effect 
approaches a limit of 


/ = _L 

R kpx 


(21-33) 


Figure 21-22 shows the results of calculations based on Eq. (21-32) as a plot of f/R 
versus local free-stream Mach number for a cylindrical body traveling with its axis at 
zero angle of attack and incident parallel rays perpendicular to the axis. The strength 
of the boundary layer as an optical element can be seen to be about the same as that of 
the hemispherical and conical shock waves previously discussed. 

The example shown in Fig. 21-22 is rather restricted. If the body under consideration 
is not cylindrical but is, for example, an ogive of circular cross section, the increasing 






848 


AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


as 

H 

o 

z 

w 

< 

o 

o 

£ 


Incident Rays 



Fig. 21-22. Ratio of focal length to radius of 
boundary-layer curvature. 


body radius causes the local flow properties to vary along the surface, and the variations 
in local free-stream Mach number and density may produce results somewhat different 
than those shown in Fig. 21-22. In addition, the curvature in two directions in this 
case would create considerable astigmatism. 

It should be emphasized that the foregoing discussion describes the behavior of the 
radiation just before it passes from the boundary layer into the protective window or 
optical device. The geometry and properties of the outermost optical element must be 
considered when the effects of the boundary layer are calculated for a particular con¬ 
figuration. For instance, if a thin protective window with the same radius of curvature 
as the boundary layer separates the boundary layer from an evacuated cavity, the 
effective focal length of the boundary layer-window combination is 

f = _L 

R fcpoo 

regardless of the flight speed or window material. 

21.3.4. Fluctuating-Boundary-Layer Effects. The turbulent boundary layer is 
characterized by random variations in velocity superimposed on the mean velocity 
distribution. Integrated aerodynamic effects of the turbulent boundary layer, such as 
heat transfer and friction drag, are fairly well known; but little is known about the 
structure of the turbulence itself, and no entirely satisfactory mathematical model is 
available. 

Several optical effects are present because of the nonuniformities in the boundary 
layer. The perturbations of the index of refraction causes the incoming rays to be re¬ 
fracted to a varying extent throughout the field; the local disturbances not being con¬ 
stant with time. When an image is focused by an optical system, these disturbances 
give rise to 












REFRACTION BY THE FIELD OF FLOW 


849 


(1) Image dancing —random shifting in apparent object position. 

(2) Scintillation —fluctuation in image intensity. 

(3) Loss in resolution —initially parallel rays not coming to focus at the same 
point in the image plane. 

Several attempts to determine the magnitude of these effects have been carried out 
[16, 17, 18, and 19]. However, variations in experimental technique and measured 
quantities make it difficult to correlate the results. Some attempt to predict the loss 
in resolution for actual flight conditions was made with the following results [20]: 

The scattering (loss in resolution) due to the turbulent boundary layer can be cor¬ 
related on the basis of a parameter (3 ', which is defined as [19] 




(21-34) 


where 8 = boundary-layer thickness 
p x = free-stream air density 

p(y) = local density as a function of distance from the wall (y) 

To evaluate this integral, some model of the boundary-layer profile must be used. 
If one assumes that no heat is transferred to the wall, that the Prandtl number = 1, 
and that 



the density ratio in air can be written 


P_ 

Px 


1 + 0.2MJ 



(21-35) 


In Fig. 21-23 the Stine and Winovich parameter (3'lp x 8 is shown. The boundary-layer 
thickness on a flat plate, 8, is approximately 

« = 0-37 <21-36) 



Fig. 21-23. Stine and Winovich scattering parameter 
versus Mach number for u/U x = (y/8) 1/9 . 










850 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


From the data of [17, 18, 19] and the analyses of [18] and [20], it appears that for the 
wind tunnel experiments, the empirical relation is 


0b « KBp x Vs (21-37) 

where K = 2.47 rad ft 5/2 /slug. The resolution limit, 0«, is defined in [19] as the angular 
diameter of the aperture at the image plane through which 85% of the total energy 
from a point source (collimated beam) passes. 

Equation (21-37) is very approximate and should be considered only as a guide to 
the general magnitude of the image degradation. The difficulties in making wind- 
tunnel measurements and the variations in technique among the various experimenters 
preclude an exact prediction of resolution loss. However, some valid general trends 
can be observed from Eq. (21-36). First, the resolution limit is proportional to the 
free-stream density (and therefore altitude). Second, the resolution loss increases 
approximately as x 2 - 5 , x being the distance from the aerodynamic leading edge of the 
vehicle. Figure 21-24 shows Eq. (21-36) evaluated for several altitudes for a range of 
Mach numbers. 



FREE-STREAM MACH NUMBER (M ) 

00 

Fig. 21-24. Resolution limit as a function of M x and body station. 


In the treatment of this problem to date, no data have been obtained on the influence 
of aperture diameter. The larger-diameter astronomical telescopes are not as suscep¬ 
tible to atmospheric turbulence [21]. Tests so far have been made with apertures of 
2 to 3 in. 





REFERENCES 


851 


21.4. Radiation from Heated Air 

Most of the infrared radiation originating in the air surrounding the vehicle comes 
from heating of the air either by its passage through a strong shock wave or by its 
being accelerated in the boundary layer. Since the radiation energy associated with 
this flow field is many orders of magnitude below the kinetic and potential energies, 
the description of the flow field can be treated independently of the radiation. Once 
the flow-field properties are established, the radiation, in principle, can be estimated 
from laboratory measurements and from theory. 

The calculation of the flow field is complex. Not only must the pressure and tempera¬ 
ture be estimated at each point in the field from the vehicle out to the bow-shock wave, 
but also the nonequilibrium and chemical kinetic effects must be accounted for. These 
two considerations are vital if realistic estimates of air radiation are to be made, 
especially at high altitudes and high speeds. In particular the molecular vibration- 
rotation relaxation times may, or may not, be long enough to provide a radiation-free 
area in the forward part of the flow field. The constituents of high-temperature air 
(in particular CO and NO which do not occur at ambient conditions) must be found to 
calculate a reasonable spectrum. 

If the fluid properties are known, the radiation characteristics can be estimated from 
standard works on gaseous radiation such as [22] and [23]. 

References 

1 . High Speed Aerodynamics and Jet Propulsion, Vol. V. Turbulent Flows and Heat Transfer, 
Princeton University Press, Princeton, N.J., 1959. 

2. Schlichting, Hermann., Boundary Layer Theory, McGraw-Hill Book Co. Inc., New York, 1955. 

3. Schneider, P. J., Conduction Heat Transfer, Addison-Wesley Publishing Co. Inc., Reading, 
Mass., 1955. 

4. Fourier, J., Analytical Theory of Heat, Dover Publications Inc., New York, 1955. 

5. Van Driest, E. R., Investigation of Laminar Boundary Layer in Compressible Fluids Using the 
Crocco Method, NACA Tech. Note 2597, Jan. 1952. 

6 . Notes For a Special Summer Program in Aerodynamic Heating of Aircraft Structures in High 
Speed Flight, Massachusetts Institute of Technology, Department of Aeronautical Engineering, 
1956. 

7. Sibulkin, M. J., Aeronaut. Sci., Vol. 19, No. 570, 1952. 

8. Equations, Tables, and Charts for Compressible Flow, NACA Tech. Rept. 1135 (1947). 

9. Stine, H. A. and K. Wanlass, Theoretical and Experimental Investigation of Aerodynamic 
Heating and Isothermal Heat Transfer Parameters on a Hemispherical Nose with Laminar 
Boundary Layer at Supersonic Mach Numbers, NACA Tech. Note 3344, 1957. 

10. Beckwith, I. E. and J. S. Gallagher, Heat Transfer and Recovery Temperatures on a Sphere 
with Laminar Transitional and Turbulent Boundary Layers at Mach Numbers of 2.00 and 4.15, 
NACA Tech. Note 4125, 1957. 

11. Beardsley, N. F., "Infrared Transmitting Windows” (U), Proc. of Infrared Information Sym¬ 
posium, Vol. 1, No. 2, Dec. 1956 (CONFIDENTIAL). 

12. Charwat, A. F., C. F. Dewey, J. N. Roos, and J. A. Hitz, "An Investigation of Separated Flows. 
Part II. Flow in the Cavity and Heat Transfer,” Journal of Aerospace Sciences, Vol. 28, No. 7, 
July 1961. 

13. Slater, J. C., and N. H. Frank, Electromagnetism, McGraw-Hill Book Co. Inc., New York, 1957. 

14. Handbook of Geophysics for Air Force Designers, AFCRC, Cambridge, Mass., 1957. 

15. Melkus, H. A., The Behavior of a Light Ray Penetrating a Supersonic Flow Field, U .S. Air Force, 
ARDC Report AFMDC-TR-59-39, Oct. 1959. 

16. Liepmann, H. W., Deflection and Diffusion of a Light Ray Passing Through a Boundary Layer, 
Report SM-14397, Douglas Aircraft Company, Santa Monica, Calif., 1952. 

17. Baskins, L. L. and L. E. Hamilton, The Effect of Boundary Layer Thickness Upon the Optical 
Transmission Characteristics of a Supersonic Turbulent Boundary Layer, Report NA1-54-756, 
Northrop Aircraft Co., Beverly Hills, Calif., Nov. 1954. 

18. Bartlett, C. J., The Scattering of Light Rays in a Supersonic Turbulent Boundary Layer. M. S. 
Thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 
June 1961. 


852 AERODYNAMIC INFLUENCES ON INFRARED SYSTEM DESIGN 


19. Stine, HA., and W. Winovich, Light Diffusion Through High-Speed Turbulent Boundary Layers, 
NACA RM A56B21, May 1956. 

20. Lorah, L. D., J. E. Nicholson, and R. E. Good, Near-Field Aerodynamic Influences on Optical 
Systems (U), Mithras Inc., Cambridge, Mass., Report MC-61-19-R-1. Sept. 1961 (CONFI¬ 
DENTIAL). 

21. Mikesell, A.H., The Scintillation of Starlight, Published by the U.S. Naval Observatory. 
Second Series, Vol. XVII, Part IV, 1955. 

22. Penner, S. S., Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley 
Publishing Co., Reading, Mass., 1959. 

23. Plass, G. N., Emissivity of the 4.3 Micron Band of Carbon Dioxide, Scientific Report No. 2, 
Aeronutronic Systems Inc., Div. of Ford Motor Co. Newport Beach, Calif. 


Chapter 22 

PHYSICAL CONSTANTS AND 
CONVERSION FACTORS 

J. A. Jenney and Richard Phillips 

The University of Michigan 


CONTENTS 


22.1. Physical Constants. 855 

22.2. Length. 855 

22.3. Area. 855 

22.4. Volume. 855 

22.5. Angle. 855 

22.6. Mass. 855 

22.7. Density. 856 

22.8. Time. 856 

22.9. Velocity and Speed. 856 

22.10. Acceleration. 856 

22.11. Force. 856 

22.12. Torque. 856 

22.13. Pressure. 856 

22.14. Temperature. 856 

22.15. Work and Energy. 857 

22.16. Power. 857 

22.17. Electrical Units. 857 

22.18. Prefixes. 857 


853 






















Tables and Charts 


Table 22-1. Physical Constants. 858 

Table 22-2. Length Conversions. 860 

Table 22-3. Equivalents of Common Fractions of Inches in Decimals 

and Millimeters. 862 

Fig. 22-1. Range Conversions. 863 

Table 22-4. Volume Conversions. 864 

Table 22-5. Angle Conversions. 865 

Fig. 22-2. Angle Conversion Chart. 866 

Table 22-6. Mass Conversions. 866 

Table 22-7. Useful Mass Units. 867 

Table 22-8. Density Conversions. 867 

Table 22-9. Time Conversions. 868 

Table 22-10. Velocity Conversions. 869 

Table 22-11. Acceleration Conversions. 870 

Table 22-12. Force Conversions . 871 

Table 22-13. Torque Conversions. 871 

Table 22-14. Pressure Conversions. 872 

Fig. 22-3. Temperature Scales. 873 

Fig. 22-4. High-Temperature Conversions . 874 

Table 22-15. Work and Energy Conversions. 875 

Fig. 22-5. Energy Conversions. 876 

Table 22-16. Spectroscopic Energy Conversions and Equivalences . 878 

Fig. 22-6. Spectroscopic Energy Conversions. 878 

Table 22-17. Power Conversions. 879 

Table 22-18. Electrical Unit Conversions. 880 

Table 22-19. Prefixes. 881 


854 



























22. Physical Constants and Conversion Factors 


22.1. Physical Constants 

Table 22-l(A) lists defined values and equivalents; Table 22-l(B) lists energy conver¬ 
sion factors; and Table 22-1(0 gives adjusted values of constants. The values are those 
recommended by the National Academy of Sciences and the National Research Council, 
as reported in Physics Today, issue of February 1964. The notation used follows that 
of this reference. 

22.2. Length (Z) 

The standard of length is the meter, defined as 1,553,164.13 wavelengths of the 
cadmium red line in air at 760 mm pressure and 15°C. Conversions from one set of 
units to another are given in Table 22-2. Equivalents of some common fractions of 
inches in decimals and millimeters are given in Table 22-3. Figure 22-1 provides a 
graphical technique for length conversions in yards, kilometers, statute miles, and 
nautical miles. 

22.3. Area (l 2 ) 

Measurements of area are based on those of length; thus there is no standard area. 

22.4. Volume (Z 3 ) 

Table 22-4 includes the common scientific units of volume. 

22.5. Angle 

An angle is usually defined in terms of a fraction of a circle, or the arc and radius of 
the arc. Given this way, it is dimensionless, although the name radian is usually 
specified. One radian is the angle subtended by an arc equal to the radius of the circle. 
When specified in degrees the angle is still dimensionless, but numerical difficulties 
arise in calculations because the number base 360 is now implicit. Conversions among 
degrees, radians, grades, etc., are given in Table 22-5. It should be noted that a mil 
is often defined in the same way as a milliradian, and is often given as 1/6400 of a circle, 
and also as a subtense (grade) of 1 part in 1000. These are all approximately equal for 
small angles. Here they are defined as follows: 

1 milliradian (1 mrad) = 0.001 radian 
1 angular mil (1 mil) = angle of 1 part in 1000 
1 military mil (1 mmil) = 1/6400 circle 

Figure 22-2 is a conversion chart for changing from milliradians to degrees and min¬ 
utes for small angles. 

22.6. Mass (m) 

Mass is the quantity of matter. Table 22-6 provides data for converting from one 
mass unit to another. The following conversion for energy and mass equivalents is 
sometimes used: 

1 g = 5.61000 x 10 26 Mev 
855 


856 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


The ratios of proton to electron mass is 1836.12, and the mass of the earth is 5.983 
x 10 24 kg (6.595 x 10 21 tons). 

Table 22-7 lists some useful mass units. 

22.7. Density ( ml ~ 3 ) 

Density is mass per unit volume. Specific gravity is the ratio of the density of a 
substance to that of water. In the metric system specific gravity and density have the 
same value; in the English system specific gravity must be multiplied by the density of 
water. Table 22-8 gives conversion factors. The density of water at 3.98°C and 760 
mm pressure is 1 g/ml; the density of water at 4°C and 760 mm pressure is 1 g cm -3 . 
This accounts for the fact that 1 liter is equal to 1000.027 cc. The density of air at STP 
is 1.293 x 10 -3 g cm -3 . Tables of the density of air as a function of pressure and tempera¬ 
ture are available in standard handbooks. 

22.8. Time (£) 

The unit of time is defined as 1/86,400 of a mean solar day. Table 22-9 gives time 
conversion factors. 

22.9. Velocity and Speed 

Linear velocity is the time rate of motion in a fixed direction; angular velocity is the 
time rate of angular motion about an axis (£ _1 ). Table 22-10 presents factors for con¬ 
version among the different units of linear and angular velocity. Mach number is also 
a measure of speed and is defined as the ratio of the given speed to the speed of sound. 

22.10. Acceleration ( lt ~ 2 ) 

Acceleration is the time rate of change of velocity in speed or direction. Table 22-11 
gives conversion factors for acceleration. The acceleration of gravity of the earth varies 
from 977.9 at the equator at sea level to 983.21 at the North Pole. Some other useful 
values are: Berlin, Germany, 981.26; London, England, 981.19; Madison, Wis., 980.35; 
New York, N.Y., 980.23; San Francisco, Calif., 979.94. 

22.11. Force ( mlt ~ 2 ) 

Force is usually defined in terms of mass and momentum. It is the quantity which 
imparts a change in momentum to a mass. Normally the famous equation f — ma is 
used, but a more critical statement is f = dpldt, where p is momentum. This allows 
for a change in mass [4,6]. Conversions are given in Table 22-12. 

22.12. Torque ( ml 2 t ~ 2 ) 

The torque about an origin of a force acting at a point is the product of the length of 
the line between the point and the origin and the component of force perpendicular to 
the line. Table 22-13 presents torque conversion factors. 

22.13. Pressure (ra/ -1 * -2 ) 

Pressure is force per unit area. One torr is the pressure of 1 mm Hg at 0°C and 
standard gravity. Table 22-14 provides pressure conversion factors. 

22.14. Temperature 

Temperature is a measure of the average translational kinetic energy of the molecules 
of a substance. The common temperature scales are Kelvin, Celsius (centigrade), 
Fahrenheit, and Rankine. The first three are given in Fig. 22-3 in a way that permits 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


857 


direct conversion. For higher temperatures, Fig. 22-4 can be used for conversion from 
Kelvin to Rankine. The governing equations are: 

5 K 

f(°F-32) = °K-273 
9 

|°C + 32 = °R- 459.69 
5 

22.15. Work and Energy ( ml 2 t ~ 2 ) 

Work is usually thought of as force acting through a distance. Energy is the capa¬ 
bility of doing work. Both have the same dimensions. Conversions among units are 
given in Table 22-15. (See also Table 22-LB.) Energy conversions for the units most 
frequently used are given in Fig. 22-5. Spectroscopic energy conversions are given in 
Table 22-16, and a detailed conversion nomograph is given in Fig. 22-6. In the two 
nomographs, the columns implying equality of energy with frequency, with wave num¬ 
ber, or with temperature are really equivalences. A dimensional proportionality factor 
h (Planck’s constant) is implied in the one case and 112k (Boltzmann’s constant) per 
degree of freedom in the other. 

22.16. Power ( ml 2 t ~ 3 ) 

Power is the time rate at which work is done. Table 22-17 contains various units for 
power and conversion factors among them. 

22.17. Electrical Units 

Table 22-18 provides conversions among several of the common systems of units 
[4-6] used in measuring electromagnetic fields, currents, electric power, etc. 

22.18. Prefixes 

Some of the useful decade prefixes are given in Table 22-19. 

References 

1. Physics Today, 17 (Feb. 1964). 

2. "Physical Constants and Conversion Factors,” General Electric Company (1955). 

3. "Range Conversion Chart,” Aerojet-General Corporation (1958). 

4. Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., Cleveland, Ohio (1958). 

5. D. C. Peasle, Elements of Atomic Physics, Prentice-Hall, New York (1955). 

6. E. R. Cohen, K. N. Crowe, and J. W. M. Dumond, Fundamental Constants of Physics, Interscience, 
New York (1957). 

7. W. A. Hiltner, Astronomical Techniques, U. of Chicago Press, Chicago (1960). 


°R = 
°C = 
°F = 


858 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


Table 22-1. Physical Constants 
A. Defined Values and Equivalents 


Meter 

(m) 

Kilogram 

(kg) 

Second 

(s) 

Degree Kelvin 

(°K) 

Unified atomic mass unit 

Mole 

(u) 

(mol) 

Standard acceleration of free fall 
Normal atmospheric pressure 
Thermochemical calorie 
International Steam Table calorie 
Liter 

(g n) 
(atm) 
(cal,/,) 
(cal/ r) 
(1) 

Inch 

Pound (avdp.) 

(in.) 

(lb) 


1 650 763.73 wavelengths in vacuo of the 
unperturbed transition 2p i0 — 5 d 5 in 86 Kr 
mass of the international kilogram at 
Sevres, France 

1/31 556 925.974 7 of the tropical year at 
12* ET, 0 January 1900 
defined in the thermodynamic scale by 
assigning 273.16 °K to the triple point of 
water (freezing point, 273.15 °K = 0 °C) 
1/12 the mass of an atom of the 12 C nuclide 
amount of substance containing same 
number of atoms as 12g of pure ,2 C 
9.806 65 m s~ 2 , 980.665 cm s -2 
101 325 N m- 2 , 1 013 250 dyn cm 2 
4.1840 J, 4.1840X10 7 erg 
4.1868 J, 4.1868X10 7 erg 
0.001 000 028 m 3 , 1 000.028 cm 3 
(recommended by CIPM, 1950) 

0.0254 m, 2.54 cm 

0.453 592 37 kg, 453.592 37 g 


B. Energy Conversion Factors 


Conversion 



Formula 

Factor 

Error 

limit 

Systeme 
International 
(MRS A) 

Centimeter-gram-second 

(CGS) 

Electron-volt 

Energy associated with 

eV 

1.60210 

7 

xlO -19 

J(eV)-‘ 

XlO -12 erg (eV) _I 

Unified atomic mass unit 

c*INe 

9.31478 

15 

10 8 

eV u _1 

10 8 

eW u _1 

Proton mass 

nipple 

9.38256 

15 

10 8 

eV 

10 8 

eV nip 1 

Neutron mass 

m n (Ale 

9.39550 

15 

10 8 

eV m„ _1 

10 8 

eW m„~ l 

Electron mass 

m e c 2 /e 

5.11006 

5 

10 5 

eV m e ~ l 

10 5 

eW m e ~ x 

Cycle 

e/h 

2.41804 

7 

10 14 

Hz(eV) -1 

10 14 

s-HeV)" 1 

Wavelength 

ch/e 

1.23981 

4 

io-« 

eV m 

lO 4 

eW cm 

Wave number 

e/ch 

8.06573 

23 

10 5 

m _1 (eV) _1 

10 3 

cm _1 (eV) _1 

°K 

elk 

1.16049 

16 

10 4 

°K(eV) -1 

10 4 

°K(eV) -1 



PHYSICAL CONSTANTS AND CONVERSION FACTORS 


859 


Table 22-1 ( Continued ). Physical Constants 
C. Adjusted Values of Constants 


Est.% _ Unit 


Constant 

Symbol 

Value 

error 

Systeme International 

Centimeter-gram-second 




limit 

(MKSA) 

(CGS) 

Speed of light in vacuum. 

.. c 

2.997925 

3 

xlO 8 m s" 1 

xlO 10 cm s" 1 

Elementary charge. 


1.60210 

7 

10 19 C 

lO 24 cm 1/2 g l/2 * 



4.80298 

20 


10 10 cm 3,2 g ,/2 8' l t 

Avogadro constant. 

.. N a 

6.02252 

28 

10 23 mol -1 

10 23 mol 1 

Electron rest mass. 


9.1091 

4 

IO 31 kg 

10- 23 g 



5.48597 

9 

10-" u 

10- 4 u 

Proton rest mass. 

... m„ 

1.67252 

8 

IO 27 kg 

io- 24 g 



1.00727663 

24 

10° u 

10° u 

Neutron rest mass. 

... m„ 

1.67482 

8 

IO 27 kg 

IO 24 g 



1.0086654 

13 

10° u 

10° u 

Faraday constant. 

... F 

9.64870 

16 

10 4 C mol -1 

10 3 cm 1/2 g ,/2 moP l * 



2.89261 

5 


10> 4 cm 3,2 g ,/2 s- , mol _, t 

Planck constant. 

... h 

6.6256 

5 

IO 34 J s 

IO" 27 erg s 


h 

1.05450 

7 

IO 34 J s 

10" 27 erg s 

Fine structure constant. 


7.29720 

10 

IO 3 . 

io- 3 


1/a 

1.370388 

19 

10 2 . 

10 2 


al2n 

1.161385 

16 

10- 3 . 

IO 3 


a 2 

5.32492 

14 

10-* . 

IO 5 

Charge to mass ratio for electron . 

... e/m e 

1.758796 

19 

10" C kg- 1 

10 7 cm 1,2 g-’ /2 * 



5.27274 

6 


10 17 cm 3/2 g _1/2 s _, t 

Quantum-charge ratio . 

... hie 

4.13556 

12 

10- 15 J sC-‘ 

10- 7 cm 3 ' 2 g>' 2 s-** 



1.37947 

4 


10- 17 cm 1 ' 2 g'' 2 t 

Compton wavelength of electron . 


2.42621 

6 

IO 12 m 

10- 10 cm 


Xc/2rr 

3.86144 

9 

10- 13 m 

10‘" cm 

Compton wavelength of proton . 

... k C ,p 

1.32140 

4 

10- 1S m 

IO" 13 cm 


Ac, p/2 

2.10307 

6 

10 ,a m 

10~ 14 cm 

Rydberg constant . 


1.0973731 

3 

10 7 m- 1 

10 5 cm 1 

Bohr radius . 

... Oo 

5.29167 

7 

10" m 

10'* cm 

Electron radius . 

... r t 

2.81777 

11 

IO 15 m 

10" 13 cm 


r*e 

7.9398 

6 

IO 30 m 2 

10- 26 cm 2 

Thomson cross section . 

... 8nr* e /3 

6.6516 

5 

lO 29 m 2 

10 ” cm 2 

Gyromagnetic ratio of proton . 

... y 

2.67519 

2 

10* rad s 'T-' 

10 4 rad s-'G -1 * 


y/2n 

4.25770 

3 

10 7 Hz T-> 

10 3 s^G 1 * 

(uncorrected for diamagnetism . 

... y 

2.67512 

2 

10* rad s 'T- 1 

10 4 rad s ’G" 1 * 

H 2 0) 







y'/2n 

4.25759 

3 

10 7 Hz T-» 

10 3 s-'G- 1 * 

Bohr magneton . 

... Mb 

9.2732 

6 

10- 24 j T-i 

10- 21 erg G "* 

Nuclear magneton . 


5.0505 

4 

lO 27 J T-‘ 

10 24 erg G _1 * 

Proton moment . 

.. Mp 

1.41049 

13 

lO 26 J T-> 

IO 23 erg G- 1 * 


Mp/M/v 

2.79276 

7 

10° . 

10° — 

(uncorrected for diamagnetism . 


2.79268 

7 

10° . 

10° — 

H 2 0) 






Anomalous electron moment corrn ... 

... (m^/Mo) — 1 

1.159615 

15 

10-* . 

IO 3 — 

Zeeman splitting constant . 

... fi B /hc 

4.66858 

4 

10 1 m-'T- 1 

10 s cm-'G 1 * 

Gas constant . 

.. R 

8.3143 

12 

10° J °K _1 mol-* 

10 7 erg °K _1 mol* 1 

Normal volume perfect gas . 

.. Vo 

2.24136 

30 

10 -2 m 3 moP 1 

10 4 cm 3 mol -1 

Boltzmann constant . 

.. k 

1.38054 

18 

lO 23 J °K-> 

IO 18 erg °K-‘ 

First radiation constant (2 nhc 1 ) . 

.. c, 

3.7405 

3 

10-‘« W m 2 

10 * erg cm 2 s _1 

Second radiation constant . 

.. c 2 

1.43879 

19 

10- 2 m°K 

10° cm °K 

Wien displacement constant . 

.. b 

2.8978 

4 

lO 3 m°K 

10 _I cm °K 

Stefan-Boltzmann constant. 


5.6697 

29 

10 * W m- 2 °K- 4 

10' s erg cm -2 s' 1 “K' 4 

Gravitational constant. 

.. G 

6.670 

15 

10 " N m 2 kg -2 

1Q-* dyn cm 2 g 2 


tBased on 3 standard deviations, applied to last digits in preceding column. C-coulomb 

*Electromagnetic system. J—joule 

tElectrostatic system. Hz—hertz 

W — watt 
N — newton 
T — tesla 
G — gauss 















































860 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


Table 22-2. Length Conversions 



m 

cm 

km 

M 

m fi 

A 

in. 

mil 

1 meter (m) = 

i 

100 

lO 3 

10 6 

10 9 

10 10 

39.3700 

3.93700 
x 10 4 

1 centimeter (cm) = 

0.01 

1 

10 5 

10 4 

10 7 

10 8 

0.393700 

3.93700 
x 10 2 

1 kilometer (km) = 

1000 

10 s 

1 

10* 

10' 2 

10 13 

3.93700 
x 10 4 

3.93700 
x 10 7 

1 micron (p.) = 

10 s 

10-" 

10* 

1 

10 3 

10 4 

3.93700 
x 10 5 

3.9370 
x lO 2 

1 millimicron (mp) = 

10 9 

10~ 7 

10-“ 

10- 3 

1 

10 

3.93700 
x 10 8 

3.9370 
x 10 s 

1 angstrom (A or A) = 

10-10 

10-" 

10 13 

10- 4 

0.1 

1 

3.93700 
x 10 9 

3.93700 
x 10 

1 inch (in.) = 

2.54 x lO 2 

2.540005 

2.54 X lO 5 

2.54 x 10 4 

2.54 x 10 7 

2.54 x 10 8 

1 

10 3 

1 mil - 

2.54 x 10 s 

2.54 x lO 3 

2.54 x lO 8 

25.400 

2.54 x 10 4 

2.54 x 10 s 

lO 3 

1 

1 foot (ft) = 

0.304006 

30.48006 

3.048006 
x 10-" 

3.048006 

X 10 s 

3.048006 
x 10 8 

3.048006 

X 10 9 

12 

1.200 
x 10- 

1 yard (yd) = 

0.91440 

91.440 

9.1440 
x 10 4 

9.1440 
x 10 s 

9.1440 
x 10* 

9.1440 
x 10 9 

36 

3.600 
x 10 4 

1 fathom (fath) = 

1.828804 

182.8804 

1.828804 
x lO 3 

1.828804 
x 10« 

1.828804 
x 10 9 

1.828804 
x 10'° 

72 

7.200 
x 10 4 

1 rod = 

5.0292 

502.92 

5.0292 

X lO 3 

5.0292 

X 10 6 

5.0292 
x 10 9 

5.0292 

X 10 10 

198 

1.98000 
x 10 5 

1 chain = 

20.1168 

2011.68 

2.01168 
x lO 2 

2.01168 

X 10 7 

2.01168 
x 10'° 

2.01168 

X 10” 

792 

7.92000 
x 10 3 

1 link = 

0.201168 

20.1168 

2.01168 
x 10 4 

2.01168 
x 10 5 

2.01168 
x 10* 

2.01168 
x 10 9 

7.9200 

7.92000 
x 10 3 

1 furlong (fur) = 

201.168 

20116.8 

0.201168 

2.01168 
x 10 8 

2.01168 
x 10“ 

2.01168 
x 10 12 

7920.0 

7.92000 
x 10 6 

1 (statute) mile (mi) = 

1.60935 

X 10 3 

1.60935 
x 10 s 

1.60935 

1.60935 

X 10 9 

1.60935 
x 10' 2 

1.60935 
x 10 13 

6.3360 

X 10 4 

6.3360 
x 10 s 

1 nautical mile (n mi) _ 
(International) 

1852 

1.8520 
x 10 s 

1.8520 

1.8520 
x 10 9 

1.8520 
x 10 12 

1.8520 
x 10 13 

7.6338 
x 10 4 

7.6338 
x 10« 

1 light year = 

9.4637 

X 10 1S 

9.4637 

X 10 17 

9.4637 

X 10' 2 

9.4637 

X 10 21 

9.4637 

X 10 24 

9.4637 
x 10 25 

3.9009 

X 10' 7 

3.9009 
x 10 19 

1 parsec = 

3.0826 

X 10'« 

3.0826 
x 10 18 

3.0826 
x 10 13 

3.0826 
x 10 22 

3.0826 

X 10 25 

3.0826 
x 10 26 

1.2707 
x 10 18 

1.2707 
x 10 20 






























PHYSICAL CONSTANTS AND CONVERSION FACTORS 


861 


Table 22-2 ( Continued ). Length Conversions 


ft .vd fath rod chain link fur mi n mi light yr parsec 


3.280833 

1.093611 

0.54681 

0.19884 

0.049710 

4.9710 

4.9710 
x lO 3 

6.2137 
x 10 4 

5.3996 
x IO 4 

1.0556 
x 10 18 

3.2438 
x IO 17 

3.280833 
x 10 2 

0.01093611 

5.4681 
x IO 2 

1.9884 
x lO 2 

4.9710 
x 10 4 

4.9710 
x lO 2 

4.9710 
x 10* 

6.2137 
x 10 8 

5.3996 
x IO* 8 

1.0566 
x IO ' 8 

3.2438 
x 10'* 

3280.833 

1093.611 

546.81 

198.84 

49.710 

4971.0 

4.9710 

0.62137 

0.53996 

1.0566 
x IO ' 3 

3.2438 
x 10" 

3.280833 
x 10 s 

1.093611 
x 10 s 

5.4681 
x 10 7 

1.9884 
x 10- 7 

4.9710 
x 10-" 

4.9710 
x lO 8 

4.9710 
x 10* 

6.2137 
x 10“ 

5.3996 
x 10-'° 

1.0566 
x 10 22 

3.2438 
x IO" 23 

3.280833 
x 10 » 

1.093611 
x 10-» 

5.4681 
x 10“ 

1.9884 
x 10-'° 

4.9710 
x 10" 

4.9710 
x 10* 

4.9710 
x IO' 2 

6.2137 
x IO' 3 

5.3996 
x IO-' 3 

1.0566 
x IO 25 

3.2438 
x 10 28 

3.280833 
x 10'° 

1.093611 
x 10 10 

5.4681 
x 10“ 

1.9884 
x 10 11 

4.9710 
x IO' 2 

4.9710 
x 10“ 

4.9710 

x 10 13 

6.2137 
x 10 " 

5.3996 
x 10" 

1.0566 
x 10 28 

3.2438 
x 10 27 

0.08333 

0.0277778 

0.018889 

5.0505 
x 10 s 

1.2626 
x IO 3 

1.2626 
x 10*' 

1.2626 
x 10~* 

1.5783 
x 10 5 

1.3715 
x 10 s 

2.6839 
x IO 18 

8.2396 
x 10~ 19 

8.3333 
x 10- 5 

2.77778 
x lO" 5 

1.8889 
x 10 = 

5.0505 

X lO" 8 

1.2626 
x IO 8 

1.2626 
x 10-* 

1.2626 
x lO 7 

1.5783 
x IO 8 

1.3715 
x IO 8 

2.6839 
x 10“ 21 

8.2396 
x 10“ 22 

1 

0.3333 

0.16667 

6.0606 
x IO 2 

0.01515 

1.5152 

1.5152 
x lO 3 

1.89394 
x IO- 

1.6458 
x IO 4 

3.2207 
x IO 17 

9.8875 
x 10 18 

3 

1 

0.5 

0.18182 

0.04545 

4.5455 

4.5455 
x IO 3 

5.6819 
x 10-" 

4.9375 
x IO 4 

9.6622 
x 10 17 

2.9663 

X IO 17 

6 

2 

1 

0.54545 

0.09091 

9.0909 

9.0909 
x IO 3 

1.1364 
x 10 3 

9.8757 
x IO 4 

1.9325 
x IO-' 8 

5.9328 
x IO 17 

16.500 

5.5000 

1.83333 

1 

0.25 

25 

0.025 

3.125 
x IO 3 

2.7156 
x IO 3 

4.2574 
x IO 18 

1.3070 
x 10-' 8 

66 

22 

11 

4 

1 

100 

10-' 

1.2500 
x IO 2 

1.0862 
x IO 2 

2.1256 
x IO 15 

6.5256 
x 10-“ 

0.66 

0.22 

0.11 

0.04 

10* 

1 

io- 3 

1.2500 
x 10 

1.0862 
x IO 4 

2.1256 
x IO 17 

6.5256 
x 10 18 

660 

220 

110 

40 

10 

10 3 

1 

0.12500 

0.10862 

2.1256 
x 10 " 

6.5256 
x 10-“ 

5280 

1760 

880 

320 

80 

8000 

8 

1 

0.868979 

1.7005 
x IO 13 

5.2205 
x 10" 

6,076.10333 

2025.37 

1012.686 

368.250 

92.0624 

9206.24 

9.20624 

1.15078 

1 

1.9569 
x IO 13 

6.0077 

X 10" 

3.1049 
x 10“ 

1.0350 
x 10 ,s 

5.1750 
x 10“ 

1.8818 
x 10“ 

4.7405 
x 10" 

4.7045 
x 10 18 

4.7045 
x 10 13 

5.8804 
x 10 12 

5.11000 
x 10 12 

1 

0.307 

1.0114 
x 10 17 

3.3713 
x 10“ 

1.6856 
x 10'* 

6.1296 
x 10“ 

1.5441 

X 10“ 

1.5441 
x 10' 7 

1.5441 
x 10" 

1.9154 
x 10 13 

1.6645 

X IO' 3 

3.25733 

1 






































862 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


Table 22-3. Equivalents of Common Fractions of 
Inches in Decimals and Millimeters 


Common 

Fractions 

Decimal 

MM 

Common 

Fractions 

Decimal 

MM 

1/64 

0.0156 

0.3969 

33/64 

.5156 

13.0969 

1/32 

.0313 

0.7938 

17/32 

0.5313 

13.4938 

3/64 

.0469 

1.1906 

35/64 

.5469 

13.8906 

1/16 

.0625 

1.5875 

9/16 

.5625 

14.2875 

5/64 

.0781 

1.9844 

37/64 

.5781 

14.6844 

3/32 

.0938 

2.3813 

19/32 

.5938 

15.0813 

7/64 

.1094 

2.7781 

39/64 

.6094 

15.4781 

1/8 

.125 

3.1750 

5/8 

.625 

15.8750 

9/64 

.1406 

3.5719 

41/64 

.6406 

16.2719 

5/32 

.1563 

3.9688 

21/32 

.6563 

16.6688 

11/64 

.1719 

4.3656 

43/64 

.6719 

17.0656 

3/16 

.1875 

4.7625 

11/16 

.6875 

17.4625 

13/64 

.2031 

5.1594 

45/64 

.7031 

17.8594 

7/32 

.2188 

5.5563 

23/32 

.7188 

18.2563 

15/64 

.2344 

5.9531 

47/64 

.7344 

18.6531 

1/4 

.250 

6.3500 

3/4 

.750 

19.0500 

17/64 

.2656 

6.7469 

49/64 

.7656 

19.4469 

9/32 

.2813 

7.1438 

25/32 

.7813 

19.8438 

19/64 

.2969 

7.5406 

51/64 

.7969 

20.2406 

5/16 

.3125 

7.9375 

13/16 

.8125 

20.6375 

21/64 

.3281 

8.3344 

53/64 

.8281 

21.0344 

11/32 

.3438 

8.7313 

27/32 

.8438 

21.4313 

23/64 

.3594 

9.1281 

55/64 

.8594 

21.8281 

3/8 

.375 

9.5250 

7/8 

.875 

22.2250 

25/64 

.3906 

9.9219 

57/64 

.8906 

22.6219 

13/32 

.4063 

10.3188 

29/32 

.9063 

23.0188 

27/64 

.4219 

10.7156 

59/64 

.9219 

23.4156 

7/16 

.4375 

11.1125 

15/16 

.9375 

23.8125 

29/64 

.4531 

11.5094 

61/64 

.9531 

24.2094 

15/32 

.4688 

11.9063 

31/32 

.9688 

24.6063 

31/64 

.4844 

12.3031 

63/64 

.9844 

25.0031 

1/2 

.500 

12.7000 

1 

1.000 

25.4000 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


863 



NAUTICAL MILES 


Fig. 22-1. Range Conversion Chart [3] 








































































Table 22-4. Volume Conversions [2,4] 


864 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


ed 

be 


cc 

s~ 

0> 


N 

o 


P5 

c 


H 








CO 

m 

CO 

o 

o 

00 

r^. 


rH 



CD 


m 

o 

o 

Q 

CO 

CO 

CD 

CO 


rH 

m 

CO 

rH 

CO 

rH 

o 

CO 

o 

o 

rH 


o 

o 


o 

d 

d 

o 


© 

o 

d 






r 








o 

o 

m 






rH 

05 

CM 

00 






CM 

rH 

rH 

m 

CM 

m 


rH 

X 

CO 

00 


CM 


00 



t> 

CD 

rH 

rH 

Tf 

rH 

o 

o 

o 

o 

CM 

o 

o 

d 


t> 

CD 

d 

d 






cm 








CO 

CM 

CO 

m 






rH 

(N 

co 

T}< 

CM 

CD 

CO 

H 

CM 

rH 

00 

00 

o 

CO 

O 

rH 


05 

o 

o 

o 

CM 




m 

© 

d 







co 

CD 

<n 

H 





in 

o 

CO 

04 

rH 

o 

CD 


in 


CM 

05 

rH 


CO 

lO 

rH 



o 

o 

rH 

o 

o 

o 


d 


05 

CM 

o 

d 

d 

rH 





h 








o 

Q0 

05 






rH 

CD 

CM 


CO 

CD 

rH 

CO 

CO 


X 

Q0 

CO 

CD 

i> 

m 

05 

rH 

CO 

CD 

CO 

in 

00 

rH 

CO 

05 

05 

r-H 

o 

CM 

O 


05 

d 


t> 

CO 

oo 

CM 

05 

d 

d 



( w > 



05 








rH 



i> 





CO 






128 

m 

CO 

CO 

m 

m 

rH 

rH 

00 

32 

16 

t> 

m 

© 

d 


CO 

CO 




05 

CO 

CM 


05 

CD 

in 

05 

m 


oo 

o 


CM 

’'t 



r-H 

CO 

r-H 

O 

oo 

o 

rH 

t> 

t> 

oo 

oo 

CO 

CM 

CM 

t— 

d 


rH 

CD 

m 

CM 



r-H 

CM 

CD 

rH 

t> 

oo 

t> 

CO 

m 

t> 

CM 

O 

o 

CD 

CM 

CO 

■'t 

in 

00 

rH 

CO 


CO 

d 

rH 

05 

CM 

o 

o 

rH 

05 


t> 

CO 

00 " 

CM 

II 

II 

II 

II 

II 

II 

II 

II 


6 


CJ 

S-. 

a> 


s 


c 

01 

o 


Xi 

3 

o 



N 


C*5 

O 


c 

C3 


• 

' — 



<D 


X 

O 


0 

3 


c 

3 


• r 

O 

/>—N 

0 

• r 

'3 

CO 

rD 

3 

• M 

3 

D 

0 

cc 





^bc 

e*3 

8 

w H3 

<x~o 

• rH 

g "■> 

Ch 

tn 3 
cC o“ 

-M ^ 

C ^ 

,© CO 

0 

• rH 

_C 

3 3 


cC 

3 

O*'—" 

aw 

bD^ 

CJ 

rH 

i-H 

rH 

rH 


















Table 22-5. Angle Conversions 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


865 


"3 

CM 

rH 

^2 
rH " 

r- 1 X 

rH 










centesim 

sec 

oo „ 

i-H O 
LO rH 
CO 

5 x 

^ - 
CO o 

00 rH 

£x 

CM m 

CD O 
CO rH 

9 X 
CO x 

CM « 

CO o 

CO rH 

9 X 
co A 

2 « 

2 ° 
O rH 

Sx 

<C 

© 

rH 

■’f 

o 

rH 

CM 

o 

rH 

iH 

*3 

CM 

rH 3 L 

^2 
i—i ^ 

^ X 

rH 










mtesim 

min 

00 - 
H O 
LO rH 

oo 

2* 

^ 7 

co A 

00 2 
o rH 

cd X 

CM „ 

CO o 

CO rH 

9 x 

CO x 

CM - 
CO o 

CO rH 

9 x 
co x 

o 

o 

o' 

"'t 

•<r 

O 

rH 

CM 

O 

rH 

rH 

CM 

1 

o 

rH 

u 











grade 

CM 

i—1 ZL 

^2 
rH ^ 

’"l X 

rH 

oo 7 

rH ' 

LO 2 
00 ^ 

rH X 

co 

S a 

oo 2 
o ^ 

cd X 

CM _ 

CO o 

CO rH 

9 X 

CO A 

CM 7 

<x> A 

co 2 

cd X 

o 

o 

CM 

o 

rH 

rH 

CM 

1 

o 

rH 

1 

o 

rH 


2 - 

v-H | 

rH O 
*-H rH 

HX 










c 

3 

H 

ca 

3 

O' 

00 ■* 

H ' 

io 2 
oo ^ 

r-5 X 

tH in 

CO A 
oo 2 
o ^ 

cd X 

CM 7 

co A 
co 2 

cd X 

CM 7 

«o A 

CO 2 

cd X 


rH 

CM 

1 

o 

rH 

rr 

1 

O 

rH 

fC 

1 

o 

rH 

circum¬ 

ference 

00 « 
t> A 

t> ^ 

cm X 

CM 2 
CO ” 

Tf X 

£ <0 

O 1 

CO o 

rH rH 

*> V 
t> x 

LO „ 

LO i 

q ° 

05 rH 

Sx 

lo 7 
q ° 

05 rH 

2x 

rH 

o 

o 

o 

LO 

CM 

o 

o « 

© A 
° 2 

CM X 

© m 

O i 

9 ° 

O rH 

9 x 

CM X 

o 7 
o A 
o 2 

CM X 

mrad 

CO * 

s® 

1-i X 

05 *7 

§2 

05 

cm X 

rH » 

®2 
00 ^ 

H X 

ed 

o 

rH 

rH 

05 

00 d 
oo ^ 

9 x 

CO 

oo „ 

o o 

t> rH 

2x 

oo - 
o o 

l> rH 

Sx 

oo 7 
o A 
t> 2 

rH X 

oo « 
o A 
t> 2 

LO " 

r-5 X 

rad 

CO 7 

r -5 X 

05 7 

o2 

05 ^ 

oi X 

rH W 

S2 
A x 

rH 

cd 

1 

O 

rH 

05 

H 

00 

oo 

CM 

cd 

oo 

o 

i> 

LO 

rH 

Oi 
t> 2 

LO ^ 

rH X 

00 7 

LO 

rH X 

oo 7 

LO 1-1 
rH X 

sec 

o 

o 

CO 

CO 

o 

CO 

rH 

LO 

co a. 

CM 2 

cd ^ 
9 x 

CM 

LO 

CO a. 

CM 2 

9 x 

CM 

o 

o 

©_ 

cd" 

05 

CN 

rH 

o 

o 

<q 

CM 

CO 

o 

CM 

CO 

o 

§2 

T}< ^ 

9 x 

CO 

o „ 

O i 

o o 

^ rH 

9 x 
co x 

min 

o 

CO 

rH 

t> c 

CO 1 

CO O 
CO rH 

co 

rH X 

LO 

-2 
CO ^ 

^ X 

CO 

LO 

t> 

CO 

cd 

o 

o 

cq 

rH 

CM 

o 

o 

id 

LO 

o „ 

O i 

2 ° 

O rH 

LO X 

O w 

§2 

Ht 

id X 

degree 

rH 

Lr « 
co i 
co o 

CO rH 
CO 

rH A 

t> 1 

t> o 

t> rH 

^ x 

CM A 

oo 

05 2 
CM rH 

*> x 

LO 

°° « 

LO 1 

05 o 

CM rH 

^ X 

LO X 

O 

co 

co 

o 

05 

o A 
© ^ 

05 X 

o « 

§2 

O 1-1 

05 X 

o? 

2 ° 

9 1 

05 X 


II 

II 

II 

II 

II 

II 

II 

II 

II 

II 





'O 




3 

-a 





ca 




0 

e 





tH 

a 

05 



c 

• pH 

H 

o 

05 

05 






C9 



s 

CO 




’d 

c 

0 

05 

tH 




_ 

o 


q- 

ca 

tH 

ca 

T3 

HH 

C 


ca 

a 

ca 

a 

05 

$ 

T3 

0 

ca 

a 

ca 

tn 

05 

H3 

ca 

feb 

• H 

co 

• ^4 

CO 

05 

& 

05 

TJ 

c 

s 

C 

O 

C5 

05 

09 

ca 

• pH 

T3 

ca 

tH 

M 

• pH 

s 

0 

o 

tH 

o 

T3 

ca 

0 

CT 1 

0 

05 

05 

0) 

-u 

C 

05 

05 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 



























MILLIRADIANS 


866 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


O 
. m 


O 


o w 

. co W 
• W 
OS 
o 
w 
Q 


o 

■ CM 


_ O 


CO 

£ 

s 

Q 

I 

J 


o 

' co 


in 

CM 


o 

CM 


(h 

CC 

X 

U 

c 

o 

• rH 

co 

Sr 

0) 

> 

c 

o 

o 

el 

< 


(N 

I 

CN 

CM 

d 

HH 

pZH 


ID 


go 

Z 

o 

So 

os 

w 

> 

Z 

o 

o 

co 

co 

< 


CD 

I 

CM 

CN 

« 

CQ 

< 


be 


be 


be 

£ 


£ * 

CO I 

£ » 

CO I 

£ « 

CO 

rH 

CO 

m 7 

to 


CM o 

CM O 

CM © 

CM 

CM o 



O rH 

O rH 

O rH 

o 

rH rH 







» X 



^ X 

rH >< 

rH X 

rH 

o 


St 

s? 

CM 

CO 

CM 

CO 

in 

CM 

CO 

© 


o 

it O 

it o 




o 

O rH 

O rH 

o 

<3 


o 

« X 
(N X 

« X 

CM A 

<N 

oi 

CM 

CM 

o 


CM 

O"} to 

OS „ 

OS 

05 




5.273 

< 10-' 

CO 1 

t> o 

CM rH 

uo x 

CO 

t> 

CM 

b 

CO 

o 

CM 

lO 

rH 

16 

32,00( 

CO 

CO ^ 

CO 

CO 







co 

^ o 

CO rH 

CM 

m 

os 

1 

o 

CO 

1 

o 

M 

1 

o 

rH 

05 « 

U0 ^ 

00 

rH 

t> 

o 





00 x 

CM 

3 X 
o 

05 

o 

9-01 

C*5 

1 

o 

rH 

rH 

CO 

o 

rH 

0.0283495 

0.453592 

907.185 





m 

CM 

in 





05 

05 

00 

1 

o 

rH 

o 

rH 

co 

O 

rH 

it 

CO 

iq 

co 

rH 

t> 

rH 




00 

m 

o 





CM 

it 

05 





m 

CM 

m 



co 


05 

o> 

rH O 


o 

o 

o 

it 

iO 

t> rH 


rH 

rH 

rH 

CO 

CO 

o 





co 

in 

05 X 





CM 

it 

o 


be 

£ 

E 

c« 

fee 


II 

II 

II 

II 

II 

ii 






"a? 


bo 




a 

'be 

£ 

£ 

ca 

Sb 

'Jj 

'C 

a5 

£ 

N 

<D 

0> 

o 

/■“N 

jQ 

T3 

C 

3 

"O 

Sr 

'3 

> 

3 

03 


c 

c 

3 

c 

So 

IS 

o 

Hi 

3 

o 

a 

s 


o 





















Table 22-7. Useful Mass Units [4,6] 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


867 



X 

04 

1 

04 

1 

•'T 

04 

1 

h 

04 

1 

■<r 

04 

1 


1 

O 

1 

O 

1 

O 

1 

o 

1 

O 

ft ^ 

rH 

rH 

rH 

rH 

rH 

X 

X 

X 

X 

X 


05 

o 


GO 

^ w 

CO 

CO 



T}< 


GO 

<M 


t> 

t> 


o 



to 

to 


rH 

GO 

GO 

GO 

GO 


05 

i-H 

T—1 

rH 

rH 






^_ s 






i' i 





r—* 

cd 





cd 

o 





o 

• pH 
rH 





’35 

6 





>> 

a> 





X 

-O 





CL 

o 





v —' 

N —^ 







c 

a 

c 

O 

c 

c 

o 

'£ 

'2 

£ 

a 

S-c 

o 

s- 

-*-> 

CO 

co 

O* 

u 

JU 

F a> 

o 

S-H 

a 

0 

0) 

a 

cfl 

cd 

s 

CO 

cd 

s 


c«-i 


Cm 




O 

o 

O 

o 

CJ 

• rH 


CO 

CO 

CO 

’£ 

s 


CO 

CO 

CO 




od 

cd 

cd 

o 

-4-0 

o 

+-> 


S 

s 


< 

< 












Table 22-9. Time Conversions (Mean Solar Time) [7] 


868 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


u 


>> 

03 

T3 


Sh 

.d 


d 

• rH 
6 


co 

d 


CO 


o 

03 

to 


s 


o 

0) 


co 


3.1689 
x 10- 8 

3.1689 
x 10 11 

03 2 

00 A 

60 O 
t— 1 rH 

CO X 

03 2 

oo A 

60 O 

rH 

co x 

CO ® 
rH A 

o 2 

03 ^ 

A X 

00 7 

o A 

^ 2 
rH ^H 

A X 

03 « 

t> A 
co 2 

<N X 

rH 

Tf “5 

? A 

10 2 
i-i ^ 

rH X 

i# <*> 

t> a 

10 2 
rH ^ 

rH X 

s 

t> i 

tO o 

rH rH 

r X 

rf 2 
i> i 

to © 

rH rH 

rH X 

^2 
03 ’” l 

60 X 

N 

60 A 
co o 

rH ^H 

A x 

rH 

<N 

Ht S’ 

<M 2 

9 x 

CO 

oo 7 
» A 

<N X 

oo *~ 
t> A 

t> 2 

C"— tH 

03 X 

00 2 
t> i 

l> o 

t> rH 

(N X 

00 2 
t> i 
t> O 
t> rH 

<n x 

t> « 

60 A 
60 2 
C£> 

rH* X 

rH 

Ht 

<N 

03 n 

lO o 

60 rH 

^ y 
00 x 

M 

60 A 
60 2 
CO ’H 

rH X 

r>. to 

60 a 

co 2 
co rH 

rH X 

hn ao 

60 A 
60 2 
co ^ 

rH* X 

t> 2 

60 1 

60 O 
CO rH 

rH X 

rH 

o 

60 

O 

Tf 

rH 

2 * 

03 O 

to rH 

X 

to A 

a 

O 

rH 

60 

O 

rH 

& 

rH 

rH 

o ® 

§2 
o ^ 

co X 

O n 

8 2 
60 ^ 

CO X 

O rj 

52 

60 

00 X 

03 _ 

60 2 
to O 
to rH 

3 x 

CO 

6 c 

O 

rH 

co 

o 

rH 

rH 

0*5 

1 

o 

rH 

o r, 

© o 

O rH 

9 x 

co A 

2 * 

2 ° 
O rH 

60 

CO X 

o ® 

60 ^ 
oo X 

03 _ 

60 2 
to © 
to rH 

o 

o 

o 

rH 

rH 

CO 

1 

O 

rH 

60 

1 

O 

rH 

© * 
o o 

O rH 

° v 

60 X 

2 « 

2 ° 
O rH 

60 

coX 

o 

§2 
rt rH 

9 x 
oo 

60 2 
to o 

to rH 

3 X 

co 

rH 

co 

1 

O 

rH 

60 

1 

o 

rH 

Oi 

1 

O 

rH 

o 

60 

o 

o 

60 

CO 

o 

o 

H* 

60 

00 

03 

10 2 
to ^ 

H X 
CO 


II 

II 

II 

II 

II 

II 

II 

II 


^ ^ 

, -V 







O 

0 ) 

CO 

£ 

cu 

CO 

ji 

'o 

a> 

co 

3 

Si 

o 




o 

03 


o 

TO 

d 

• pH 




a> 

d 

o 

d 

£ 




C 0 

o 

CJ 

o 


t . 


V 

T3 

d 

o 

03 

Qj 

o 

at 

CO 

• rH 

• H 

H 

03 

CO 

O 

Sh 

O 

• pH 

o 

03 

CO 

O 

d 

CD 

03 

d 

d 

• pH 

3 

pH 

a 

SH 

d 

o 

CD 

£ 

Sh 

CD 

03 

CO 

£ 

E 

d 

E 

A 

T 3 

>> 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 


co 

d 

o 


co 

u 

03 

> 

d 

o 

o 

CO 

3 

o 

03 

c 

68 


03 

cj 

CO 




3 

O 

A 

Sh 

J8 

'o 

co 

d 

c8 

03 

£ 

Tf 

rH 

60 

CO 

t> 

CO 


CO 

>> 

CD 

T3 

Sh 

« 

'o 

CO 

d 

CD 

03 

6 

60 

lO 

(M 

tO 

60 

CO 

II 

Sh 

CD 

03 

>> 

Id 

a> 

• rH 

co 


co 

>> 

CD 

T3 

Sh 

JD 

O 

CO 

d 

CD 

03 

£ 


CO 

>> 

CD 

T3 

s- 

cD 

i 1 < 

o 

CO 

d 

CD 

a> 


tO co 

60 co 

co co 


















Table 22-10. Velocity Conversions 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


869 


I 

Z 

s 

c 


X 

5. 

E 


i 

s- 

X 


C 


i 

t. 


J2 


£ 


£ 

c 


£ 


8 


CO 


£ 


I 



I 



00 ** 
s= 

05 ^ 
H X 

00 

CO 

H< 

05 

rH 

00 „ 

CO o 

Tf 1 ~H 

2 x 

CM i 

w 2 

2 X 

^2 

05 ^ 

in X 

Tf 

CM 2L 

00 ® 

CM ^ 

2 X 

CO 

go 

CO 

Z* X 

CO 

os 7 

10 2 

03 2 

CO ^ 

in X 

1.37056 

x 10 s 

co 7 
o 

rH 

CD 

2 X 

05 7 

oo L. 

00 o 
co ^ 

2 X 

rH 

05 « 

9© 

CO II 
(M ^ 

CM X 

05 

CO 

CO 

CM 

CM 

03 el 

CO o 

CO rH 

X 

CM X 

g 2 
in X 

h< _ 

O i 

00 o 

rH rH 

s* 

§ 2 , 

CO 

Z^ X 

CO 

00 „ 

CM O 

H* rH 

3* 

00 _ 

^ 1 
<Ji o 

O rH 

<=1 x 

CD X 

CN i 

£2 

Sx 

s - 

<J) 1 

CO o 

<Ji rH 

oo 

®x 

rH 

00 

o 

o 

m 

rH 

rH 

>—i N 

H O 
GO rH 

n X 

r 1 ■*• 
r O 

GO rH 

rH „ 
i-H O 
00 rH 

hX 

§2 
o ^ 

CO X 

I s 

CD 

coX 

GO „ 

O O 

O rH 

Sx 

H 1 „ 

t> o 

00 rH 

2X 

CO - 
00 o 

rH 

£x 

CO 

CO 

co r: 
CO ^ 
00 X 

rH 

o 

00 

CM 

m 

CD 

O SL 

CO ® 
t> ^ 

2 x 

CD 

CM 

^ X 

rH 

CM 

X 

rH 

CM 

£© 
2 ^ 

^ X 

rH 

I s 

CO 

coX 

2 * 
o o 

CM rH 

5- 

CO 

®2 

00 ^ 

2 X 

CM 

00 

00 a. 

h* 2 

CM ^ 
2 X 

CM 

O r 

t> o 

CO rH 

°i X 

CO A 

rH 

CM 

rH 

o , 

CD O 

CO rH 

2 x 

CD X 

CO r 

05 rH 

2 x 
o A 

o ~ 

§2 
© rH 

CO X 

o 

o 

o 

CO 

CO 

2 « 

2 ° 
O rH 

CO 

Z- X 
CO 

° 7 

5 ® 

rH r ”' 

oi X 

GO 

CM 

in 

05 

o 

rH 

CD „ 

CO o 

05 rH 

2 x 
m x 

t> «-i 

2 X 
CD 

rH 

2 « 

O i 

2 ° 
■^f rH 

*® X 

CM X 

o 7 

*2 

CO X 

CO 
. 05 

o 

2 

rH 

05 

rH 

m 

2 

rH 

m x 

05 O 

153 2 
CO ^ 

2 X 

CQ H* 

8 2 
CO ^ 

2 X 

CO 7 

153 2 

03 2 
CO ^ 

in X 

m « 

H ' 

t" 2 

CO ^ 

r-i X 

00 H 

w 2 

-t ® 

CO rH 

rH X 

00 7 

03 2 
00 2 
<x> ^ 

od X 

rH 

oo 7 

00 o 

05 rH 
Tf 

H X 

—' * 

° 2 
00 ^ 

CO X 

CM * 

in i 
oo o 

CD rH 

3* 

CM 7 

-2 

Tf —• 

2 X 

05 7 

So 

2 X 

So 

CM ^ 

CO X 

i> 7 

2 o 

CM ^ 

CO x 

co 7 

rH O 
CM rH 

<© x 

CO » 

So 

m ^ 

2 X 

° 7 

© 2 

00 ^ 

rH X 

rH 

00 

Cr- 

o 

m 

rH 

rH 

CD r 

00 i 

Tf o 

CM rH 

^X 

^ Oi 

rH i 
rH O 

GO r-H 

3x 

t> ® 

CO I 

t> o 

lO rH 

2 x 
m x 

05 7 

S O 

2 x 

7 

®2 
rH ^ 

CO X 

00 •» 

© i. 

GO 2 
CM * H 

CO X 

00 

o 

00 

CM 

CO 

00 

© 2 > 
00 rH 

2 X 
CO x 

CM *» 

w® 

CO ^ 

00 x 

rH 

© 

00 

CM 

m 

CO 

CO 

o 

rH 

2 

[- 

©_ 

CD 

CM 7 

°2 

rH 

05 X 

si 

CO 

2 X 

00 i 

m o 

t" rH 

2 x 

CM X 

<x> 

CO 

rH 

00 

t> 

00 

2 

rH 

Q 

05 ^ 

CO x 

o _ 
t> o 

CO rH 

S X 

CO 

° X. 
H o 

CO rH 

£x 

CO 

rH 

<N 

rH 

O „ 

CO o 

CO rH 

CD X 

”2 
05 ^ 

2 X 

t> 

00 2. 

CM 2 

2 x 

rH 

05 ■» 

So 
2 x 

rH 

rH 

”2 
CO ^ 

2 x 

oo 7 

00 Jr. 

^ 2 

rH X 

C" _ 

05 i 

CO O 

CM rH 
2 x 

CM X 

m 

1 

O 

rH 

CO 

1 

o 

rH 

rH 

in 

2 2 
N x 

° 7 

00 2 
h 2 
o 1-1 

CO X 

CO 

05 

o 

q 

rH 

m 

05 

rH 

m 

« 

rH 

00 7 

So 

tT ^ 

2 x 

m 7 
o o 

in rH 

2 x 

s * 

CD i 

O O 

CD rH 

Tf 

2 X 

CM 7 

50 2 

Hf 

Tf X 

c~ 7 
o 1. 

2 
rH ” 

2 x 

CM 

1 

O 

rH 

rH 

© 

rH 

CM 

S2 

'"x 

n 
© ^ 

2 x 

©S 

O rH 
CO 

® x 

m 

2 o 
in 1-1 

°q x 

rH 

oo 7 

S o 

CM X 

m * 

9 ° 

in rH 

2 x 
t- A 

S7 

o o 

CD rH 

Tf 

2 x 

CM 7 
t" 1. 

CO 2 

Tf X 

o g 

2 X 

rH 

© 

rH 

m 

O 

rH 

m 

CM 

© _ 

00 o 

rH 

2 X 

CO o 

05 O 
O rH 

CD 

® x 

m 

2 o 
m 1-1 

°o X 

rH 

00 _ 
m o 

O i—t 

x 

CM X 

CO „ 

m i 
o o 

in rH 
2 x 

9 o 

CD rH 

Tf 

2 x 

CM - 
o O 

CD rH 

Tf 

2 x 

r> _ 
o o 

Tf rH 

2 x 


O 

0) 

co 


-T3 

C 

O 

o 

0) 

CO 

Sh 

8 . 

In 

a 

05 

£ 


c 

01 

o 


o 

05 

03 


~o 

c 


05 

03 

(h 

8 . 

u 

01 

£ 


II II II 


V 

0) 



CO 



£ 



'O 

8 

co 

1 

o 

<z> 

c 

o 

o 

a> 

03 

C 

eft 

g 


T3 

u 

C 

C 

a 

u 

o 

1 

eft 

O 

S 

eft 

05 

H-> 

05 

£ 

In 

8. 

S-I 

8. 

O 

X, 

U 

c 

>4-» 

^5 

cl 

rH 

rH 

rH 





























Table 22-11. Acceleration Conversions 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


870 


L 

J3 


6 


i 



-C 


E 


i 

c 

E 

£ 


M 

I 


£ 



I 

c 
• ^ 

£ 

42 


N 



-C 

c 


c 

£ 

c 


s 

co 


u 


8 
£ 05 


N 

I 

U 

-C 

£ 


CM 

I 


c 


£ 

£ 


£ 


Tf 

05 

co 

co 

IM 

<M 

St 

CO o 

i-H 

*! x 
CD X 

T - 
© 1 

CD O 
<M hH 

^x 

2 ^ 
t> 1 

CO o 

»-H rH 

9 x 

CD X 

CO „ 

CD i 
© © 
© r 

^ X 
ID 

05 *o 
<N ' 

00 O 
r> ^ 

9 x 

rH 

4.38413 

x lO 9 

6.81820 

x 10 -' 

05 i 

CO o 
05 i-H 
X 

“ x 

Cl 

X i 

X o 

X rH 
rH 

r-i X 

2 2 
© © 
© rH 
© 

X 

© 

rH 

2.77778 

x 10- 4 

rH 

8.05299 

X 103 

05 

CD 

CO 

CM 

Cm’ 

St 

CO o 

*-H rH 

9 x 

CO X 

Tf 

05 

CD 

CO 

(N 

© 

H* SC, 

CD 2 

9 x 

(N 

2 « 

00 i 

s ° 

00 rH 
CD 

CD X 

s * 

<M i 

X o 

C'* 1 -H 

^x 

to 

i£S 

© r 
^ X 

<N 

o _ 

(M i 

X o 

1 -H 1 -H 

X 

cb X 

CM 

S 2 

9 x 

if 

© 

© * 
© © 
© rH 

9 x 

rH 

CD 

rZ X 

X 

rH 

© 

§2 
© ^ 

9 x 
© 

05 

c O 

CO 

CM 

CM 

St 

CO o 

1 -H 1 -H 

9 X 
<x> x 

X *- 

© i 

CD o 

CM rH 

^X 

CO ^ 
l> 1 

co o 

1 -H rH 

^ X 
CD * 

CO N 

CD 

r © 

00 r 

9 x 

CD X 

St 

00 o 

t> rH 

Sx 

2 » 
rH 

Tf © 

© rH 

2 x 

© _ 

(M i 

CD © 

rH rH 

« X 
© X 

S 7 
x o 

05 rH 

2 x 

£ « 

CD i 

© © 
© rH 
rH 

r 4 X 

© n 
© © 
© rH 
© 

rA X 
© 

rH 

s - 

t> 1 

t> o 

t> rH 

^7 X 

CM X 

i-H 

St 

CO o 

*“H H 

^ X 
co A 

S r- 

O i 

CD O 

CM rH 

^x 

T - 
© 7 

S o 

05 ^ 
t> 

rf X 

lO 

CO *r 

§2 
(M ^ 

«> X 

rH 

CQ i 

00 o 

t"”- rH 

2 x 

2 t 

S 2 

^x 

rH M 

S “ 

2 2 
<M 

rH X 

05 i 

CO o 

05 rH 
X 

rH X 

ig« 

© i 
© © 
© rH 

9 x 
© x 

t> » 

© 7 
© © 
© rH 

^ X 
© 

rH 

s - 

p- 1 

fz ° 

t> rH 

^7 x 

CM A 

to X! 

O 1 

CD O 

rH rH 

*7 x 

t> A 

2 - 

1 

O 

rH 

^7 X 

CM A 

tO ^ 

00 o 

CO rH 

Sx 

St 

00 o 

CD rH 
Tf 

5 x 

S 1/5 

05 i 

00 o 

1 -H 1 -H 

2 x 

g - 

O 1 

00 o 

CD rH 

iri X 

© 

© 

© 

© 

© 

ID 

a « 

00 i 

CC o 

X rH 

2 x 

St 

X o 

ID rH 

X 

X 

X 

o 

§2 
© ^ 

9 x 
© 

© T 

S 2 
2 x 

rH 

© 

© s, 
00 2 

9 X 
© 

© - 
© © 
© rH 

£x 

CD 

^ i 

o 

rH 

tH 

M X 

© 

© 2 
© ^ 

9 x 

CD 

o 

rH JL 

’—i 2 

00 ^ 

9 X 

rH 

CO 

00 

o 

00 

CM 

CO 

”t 

CO o 

1 -H rH 

Sx 

CO 

00 

o 

00 

<N 

co 

© 

© 2 
© ^ 

9 x 

CO 

St 

CO O 

CO rH 

« x 

00 A 

£ 

X 1 

Tf O 

1 -H rH 

CVJ A 

© 

ga> 
© ^ 
9 x 
© 

rH 

8 

© 

g© 
© ^ 

9 x 

rH 

© 

gS 

CD 

9 x 
© 

fH 

© 

© 

© 

Tf 

rH 

© 

© 2 

9 x 
© 

CO 

00 

o 

00 

CM 

CO 

CO „ 

^ 7 
co o 

i-H i— t 

S x 

i-H 

-2 
co ^ 
ID x 
(M 

2 * 

Hf 1 

co o 

rH i-H 

2 * 

2 * 

CO I 

CO o 

CO rH 

« x 
oo A 

r? cd 

X i 

Tf o 

*-H rH 

« X 

M x 

O i 

O o 

X i-H 

© x 

rH 

St 

S 2 

^ X 

© 7 
© © 
© rH 
© 

r4 X 

© 

gb 

CD 1-1 

9 x 
© 

CD 

CD 

CD 

Tf 

rH 

t> - 

o 7 
o 

L- rH 

Sx 

CD 

CD 

CD 

rH 

5.10235 

X 10* 

IM 

2 <-• 
T X 

i-H 

O 

2 o 

co ^ 
® X 
co 

IN 

i-H ^ 

'T x 

rH 

© 

© s, 

© 2 

9 X 

rH 

© 

© ^ 

9 x 

CD 

rH 

© 

<M 2- 
© 2 
© ^ 

9 x 

rH 

© 

© 2 

9 x 

Tf 

M © 

© rH 

9 x 

<M X 

© 

© - 

rH ^ 

<M X 

00 

© 

§2 

CD ^ 

9 x 

(M 

© 

So 
© — 

9 x 

© 

© 

go 
ao ^ 

9 x 

(M 

1.41732 

X 10 5 

o 

S® 

co ^ 
^ X 
co 

r7! M 

CD 1 

CO o 

05 rH 

2 x 

o _ 
f~ o 

CO -H 

^x 

CO 

2 « 

© © 

© »-H 
© 

Z- X 
© 

rH 

St 

O 

t> rH 

^x 

© 

§2 
M ^ 

9 x 

rt 

(N 

rH 

© 

§2 
© ^ 

9 x 

Ch 

© 

§2 
CD —I 

9 x 

IM 

© 

© X 

© 2 
© ^ 

9 x 
© 

© 

§2 

CD ^ 

*> X 

rH 

© 

O 3- 
© 2 

9 x 

CD 

o _ 
t> o 

CO rH 

£x 

CO 

1-H *4 

CD i 
co O 

05 *-H 

2 X 

rH er . 

00 i 

l> o 

CO rH 

9 x 
co x 

nj ci 

CD i 

co o 

05 1 -H 

2 x 

rH 

® , 

I s - | 

t- o 

t> rH 

^ X 

(N X 

© 7 
© © 
rH f-i 

^ x 

[H X 

(M 

rH 

CO i 

CO o 

X rH 

^x 

X 

n 
© ^ 

M X 

O 

§2 
© ^ 

9 x 
© 

© 

§2 

CD ^ 

^ X 

rH 

O CD 

00 i 

CC o 

CC rH 

CC 

2 X 

iT 

© 

§2 

CD ^ 

9 X 

rH 

o „ 

o£> 

O 1 -H 

CD 

£ x 

i-H 

St 

S 2 

^x 

rH 

© 

^2 

Ht - 1 

^ X 
© 

© 7 

o Z, 
n* 2 

iM X 

<C 

lO i 

o 

to rH 

9 x 

t> A 

X 

gj & 

9 x 

rH 

© _ 

© 1 
© © 

Tf rH 

°x 

X 

© 

So 

M ^ 

9 x 

rH 

•<f 

© ^ 

9 x 
© 

Tf 

© 2 - 

© 2 

9 x 

rH 

o ^ 

1 

2 ° 

L- i-H 

^ x 

H* 

© 2 , 

© 2 
© — 

9 x 

rH 

o 

82 

05 

9 X 

rH 

§ 2 > 

CD 

X 

CO 

rH 

o 

o 2 
o ^ 

9 X 

CO 

Tf 

So 

© 

9 X 
co 

© 

® 2 
hJ< r 

*1 X 
© 

St 

2 ° 
rH 

« X 
M X 

X 

o © 

<N O 

O rH 

05 X 

CO 

© 

S & 
© - 1 
9 x 

rH 

© 

© H 

w 2 
© ^ 

9 x 

© 

i-H 

x 2 

05 O 

rH rH 

*> V 

CM 

So 

X 22 
to ^ 

^ X 
t> 

© 

M 2, 

00 2 

9 x 

IM 

So 
© 1-1 

9 x 

go 

CD 

Z- X 

CO 

rH 

s - 

C- 1 

o 

t> rH 

^x 

rH 

© 

^2 

Hf •“* 

X 

© 

O „ 

© 1 

O © 

»—1 

« x 

M X 

© » 

© 7 

© © 
© rH 

9 x 

tr X 

© 

05 —• 

9 x 

rH 

© _ 

© i 

© o 

■^f rH 

Sx 

© 

00 D5 
© 2 

9 x 

rH 

Tf 

S O 

X 

05 ^ 

^ X 

to 

So 

© 

9 x 

i-H 

o _ 

^ 1 
o o 

L- rH 

Tf 

h: x 

So 

© 

9 x 

rH 

1-H 

2 - 

t> i 

£ ° 
t''* 1 -H 

£ x 

oT 

CD © 
r r —1 

«> X 

o x 

2 - 
L- 1 

c- o 

l> rH 

x 

Cl 

O © 

H* rH 

9 x 

IM X 

© 

© ® 

© e, 

10 2 

9 x 

Ch 

00 ® 

X 7 

05 o 

IO rH 

2 x 

© _ 

© i 
© © 
If rH 

S X 

CD 

© T 

S 2 

5 x 

© 7 

82 

9 x 
© X 

■Cf 

© 2 - 

© 2 
© ^ 

9 x 

rH 

o _ 

1 

s ° 

rH 

Tf 

^ x 

CC 

i> 7 
^ 2 

^ X 

i-H 

o _ 

1 

2 ° 

L- rH 

Tf 

5 x 


TJ 

c 


co 

u 

8 . 

"3 

C 


CO 

8.f 

5 $ 

4> e 


3 

C 

£ 

u 

8. 

3 

D 

C 


u 

3 

O 

X 


u 

D 

O 

-C 


8. c 8.J 

S'E 


05 c 

E 5 


U U 
05 C 

BS 


T3 

C 

o 

c-> 

o> 

CO 


T5 

c 

o 


o — 

§ 7 

f 8 

&T* 

u Sh 

CD jC 

IB 


T3 

C 

o 


SU 

CO 

■C 

CJ C 
C •- 


5 

3 

C 


8. a 8. 8- 


3 

3 
C 

~ £ 


k» c 

8 -a 

JS 

y c 
c •- 


H 

3 

O 

JC 

Im 

8 . 

u 

3 

O 

J5 ^ 

u 9 

-C 

o c 
C •- 


T3 

C 


co 

I* 

& 

"O 

C 


M , 

U * 

8 . 


3 

C 


8 . 

3 

3 

C 

Ejt 

t- L 

&1 
8 a 


ii ii 


T3 

C 

XJ 

c 

o 

8 

0 ) 

CO 

u 

U 

8 . 

8 . 

3~ 

'O 

c 

minu 
1 sec 

8 

05 

05 r-H, 

u 7 

H 

0> CJ 

a 05 

05 

8 9 

c; w 

lB 


































Table 22-12. Force Conversion* [4,6] 
dyne newton g wt kg wt poundal lb wt 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


871 


1—« 

So 

rH ^ 

rH 

cm® 

rH ^ 

co « 

CM i 

o o 

»—< i—( 

CO N 

CM i 

o o 

rH I—1 

CM « 

?2 
lO ^ 

lO 

o 

o 

o 

rH 

i— i X 

rH X 

d X 

d x 

r-i X 

d 






rH 



1—1 ® 

rH 

CD « 

CD 

X 



00 A 

oo 

Tt ' 


o 


o 



O ® 

o 

rH 


o 

CM ^ 

CM 

CM 

CM rH 

CM 

CO 


o 

cm X 

CM X 

CM 

o 


CM 





o 



° * 

00 zi 

CM 1-1 

O 

CO 

00 

CM 

CM 

CO 

05 

O 

t> 

CM 

CO 

o 

05 

rH 

"t 

rH 

X 

Tf 

lO 

X 

t> 

o 

o 

o 


CM 

CO 

CD 

<o 

ovl 

l> 

05 

rH 


X 

05 

05 

LO 

o 

X 

id o 

rH 

o 



X 

rH 

o tH 
th X 

o 

rH 

o 

o 

rH 

V 

rH 

o 

lO 

t> 

o 


o 



o 

o 

05 








rH 

o 

rH 

05 

rH 

o 

o 

o 

X 

05 

o 

05 

LO 

CO 

o 

X 

rH 

t> 

o 

o 


rH 


LO 

o 

o 

rH 



rH 

Hf 

05 

o 









lO 







CD 

CO 

LO 

CD 

lO 

CM 

CM 

X 

X 

lO 


o 

CD 

X 

CD 

o 

rH 

00 

O 

X 


05 



05 

00 

rH 


X 



o 

05 

d 


X 



o 






in 

LO 

CD 

CO 

LO 

CO 

CD 

x „ 

CM O 

CM 

X o 

-*t X 

CO © 


o 

X —t 

^ I-t 

rH 


rH 

o 

o 

X w 


x w 



00 

05 

00 

05 

H X 

Tt X 

00 x 

II 

II 

II 

II 

II 

II 

II 





X 

bn 



-M> 

X 

bp 

• pH 

05 

* 



'a> 

6 

dyne 

newton 

gram w 
(g wt) 

kilograi 
(kg wt) 


CO 

X 

G 

3 


a 



X ^ 

box 

'S Z 

> o 

> o 
C o 
o CM 


X 

X 

CD 

© 

00 

05 

Cm 

o 

bo 


CO 

05 

j3 

> 

05 

X 

-u> 

C 

o 

X 

05 

CO 

cO 

X 

05 

Li 

cO 

05 

C5 

U 

a 

Cm 

o 

CO 


C 

3 

05 


O 

CO 

X 

CO 

X 

C 

CO 

3 

C 

o 


CO 


fc 

O 

00 

os 

w 

> 

£ 

O 

U 

w 

d 

O' 

os 

o 

E-i 


co 


CM 

CM 

W 

>5 

00 

< 

E- 1 



<o 





d 

o 





r— . 

rH 

CM 

■<* 

rH 


3 

X 

O 


rH 


X 

CM 

X 

CM 


X 

rH 

G 

3 

a 

rH 

X 

t> 

X 

CM 

X 

X 

CM 



CM 






t» 






o 





c 

rH 

o> 



05 

• pH 

X 



05 


rH 

t> 

CM 

rH 

rH 

CM 

t> 


rH 

CD 



X 

X 

lO 

X 

X 



o 


CO 






00 






o 





d 

rH 

lO 


X 

X 



O 


X 

o 


X 

X 


X 

rH 

£ 

r>. 

X 

rH 

X 

X 


LO 

CM 


o 

o 


t> 

t> 


o 

o 


X 












00 




n 


© 




© 

£ 

rH 

X 


X 

X 

CM 

X 

rH 

X 

be 

t> 

rH 

X 

rH 

o 

X 

05 



o 

o 


rH 


d 

o 

05 


o 




CM 


rH 








r- 

X 

X 



o 

o 

O 

o 

g 


rH 

rH 

rH 

rH 

C5 


X 

X 

X 

X 

05 

rH 

X 

X 

X 

o 

G 


X 

X 

05 

Tf 

>> 


o 

X 

CM 

rH 



X 

X 

rH 

CM 



05 

rH 

rH 



£ 

o 

05 

G 

x 

Sh 

05 

-u 

05 


G 

05 

CJ 

05 

G 

Jo 

X 


G 

pH 




• pH 


HH 

£ 

£ 

E 

X 

X 

be 


w 

X 

+-> 

X 

Q 

u, 

<v 

w 

3 

G 

• PH 

i 1 

05 

X 

♦ ’ 
X 

E 

be 

• pH 

be 

• pH 

E 

CO 

&D 

05 

£ 

05 

£ 

X 

G 

X 

G 

jo 

3 

3 

• pH 

X 

a 

a 

rH 

rH 

rH 


CO 

X 

C 

3 

a 


o 

,o 


-a 

G 

3 

a 

























Table 22-14. Pressure Conversions [4,6] 


872 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


M 

I 



c 

o 


e* 

I 


c 

c 


a 


c* 

I 





N 

I 

d 
• ** 

-C 


s 

3 


be 

K 

£ 

£ 


bo 


£ 

cd 

be 


<u 

c 

>> 

H3 


CO 50 

i—i « 

t-H T 

CO « 

rH 

o 

H* 



3 © 

5 o 

S 2 

S° 

oo 

m 

CM 

m2 

Tf 

rH 

© ^ 

o ^ 

o 1 

CO T " H 

o 

O 


rH 


r4 X 

i-H X 

H X 

i-H X 

rH 

o 

X 







o 




Tf 

CM ® 

*0 O 
CM 2 

CM 1 

CM 7 

_ <0 

GO i 

oo 

■'t 

1 

CO ® 

S 2 


Tf 

T»« 

rH O 
rH 

rH O 
rH H 

CD O 

CO rH 

CO 

t> 

o 

^2 

rH 

Tf 

CD 

O 

d X 

X 

•> X 

00 x 

o 

X 

cd X 


O 





o 




d 

CD 









00 

CO 

O 

CM 

O 

O 

2.0482 

0.20482 

2.7845 

21,162 

144 

rH 

288,000 

2,000 

o 









© ' 

CO 

CM 

CM 

Tf 

CO 

(M 

CM 

h* 

CO 

CO 

05 

CD 

05 

CD 

rH 

H* 

CD 

0 

0 

0 

05 

00 

00 

rH X 

rH 

o 

o 

o 

rH 

o 

rH 


O 

O 

cm" 

cd 

rH 


o 

o 

o 



d 



cm *r 


Tf l» 

m 

rH 


CO 

Tf 


CM 

05 

O 

<x> 

0 

o> <L 

CD 2 

® o 


CO 

rH 


o 

00 

m 

CM 

m 

00 ^ 

co ’ —l 

CO 1—1 

o 


CD 

t> 

CD 

05 

d 

d X 

05 X 

05 X 

o 

o 


© 

d 


00 

rH 

co ^ 

S 1 

©2 
lO ^ 

X 

0.73556 

0.073556 

rH 

760 

51.715 

0.35713 

103,430 

71,826 

l> 

05 



in 

CM <r 



0 

0 

00 




05 

CO O 

o 

CM 

CM 

Tf 

o 

o 


m 

CO 1-H 

cd 

o 

t> 

00 

CD 

CD 

rH 

o 

rH 


CO 

rH 

Sx 

00 

Tf 

O 

■<* 

t> 

oT 

o 







rH 


t> 









05 

rH 

o 


rH 

m 

05 

m 

CO 

CM 

CO 

t> 

o 

CO 

o 

CM 

00 

0 

CM 

CD 

00 

Tf 

rH 

o 

rH 

o 

CO 

o 

00 

<d 

CD 

r- 

o 



rH 

rH 

t> 

d 

rH 

05 

d 










iO 

CO 

lO 

CO 

CM 

CM 

iO 


0 

C7> ao 

O „ 

rH 

CD 

d 

00 

05 

CD 

o 

00 

05 

cd 

CO 

CO 

i-H 

Id o 

00 d 
9 x 

rH 

Tf 

<35_ 

00” 

CD 

00 

ao 

00 0 

r- i-H 

2x 

CD O 
t> i-H 

« x 
05 X 

1! 

II 

II 

II 

II 

II 

II 

II 

II 


a) 

h-> 

O) 

£ 

c 

<x> 

cd 

O) 

u 

cO 

3 ^ 
o*d 

CO I 

£ 

a ° 
0) 
® c 
£ >» 
TJ d 


a> 

CD 

£ 

• ^ 

c 

CD 

CD 

CD 

>H 

CO 

3 

cr 

cn 

Sm 

05 : 

ex 


£ 

CD 


£ 


(h 

a; 

-U 

CD 

£ 

CD 

bn 

3 

3 

O' 

co 


>> 

u 

3 

cd 

s- 

0) 

£ 


41 

ex 

0 

s* 


CD 

S-i 


OJ 


05 

£ d 

H-> 

CD 

bo 


3 1 

tc 

ex 


£ 

• 

c 

tn 

0 3 

£ be 

• r-H 

c 

c 

£ £ 

£ 6 

£ 

£ 

CO w' 

rH 

rH 


rH 


M 

CD 

C 

CD 

In 

3 

3 

cr 

co 


1.5 

is 


£ 

3 

3 

a* 

CO 


& 9 &. 


o £ 

CX w 


JS 

cd 

c 

a> 

C 

3 

3 

0*JT 

CO | 

u* d 

&- £ 

r- C 

S3 


J 

3 

Sm 

3 

3 

3* 

CO 


Sm 1 

C § 
o -*j 



















380 

370 

360- 

350 

340 

330 

320 

310 

300 

290 

280 

270 

260 

250 

240 

230 

220 

210 

200 

190 

180 


HYSICAL CONSTANTS AND CONVERSION FACTORS 


873 


JS Degrees 
rade Fahrenheit 


Degrees 

Kelvin 


Degrees 

Centigrade 


Degrees 

Fahrenheit 


230 


170 



-100 


-148 

Water 010 
Boils 


160 

— 


-110 


-166 

194 


150 



-120 

- 

-184 

176 


140 



-130 


-200 

158 


130 



-140 


-220 

140 


120 



-150 


-238 

122 


110 



-160 


-256 

104 

- 

100 



-170 


-274 

86 


90 



-180 


-292 

68 

— 

80 



-190 

— 

-310 

50 


70 



-200 

— 

-328 

I ce 39 
Melts 


60 



-210 


-346 

14 


50 



-220 


-364 

-4 

— 

40 

— 


-230 

— 

-382 

-22 

- 

30 



-240 


-400 

-40 

— 

20 



-250 

— 

-418 

-58 

— 

10 



-260 


-436 

-76 

- 

Absolute n 
Zero 



-270 

-273 

kl6 

-454 

-459 

-94 

— 

°C = 

f(°F 

- 32) 



-112 

— 

°K = 

oc + 

273 



-130 

— 







-148 

— 








Fig. 22-3. Temperature Scales 








874 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


Degrees 

Kelvin 

7,000 r 


6,000 


5,000 


4,000 


3,000 


2,000 


1,000 


0 


Degrees 

Rankine 

F 12,600 

E - 12,000 
^ 11,000 
- 10,000 
j- 9,000 
E- 8,000 
^ 7,000 

- 6,000 

E- 5,000 

- 4,000 
E- 3,000 
E- 2,000 

- 1,000 

- 0 


Fig. 22-4. High-Temperature 
Conversions [3] 





PHYSICAL CONSTANTS AND CONVERSION FACTORS 


875 


<N 

w 

2 

O 


w 

> 

2 

O 

O 

o 

Oh 

w 

2 

W 

Q 

2 

<C 

sc 

oi 

O 

£ 


id 


CQ 

CQ 

w 

►J 

CQ 

< 


kg m 

0.10197 

886I0I0 

8-01 X 

U610'T 

107.56 

3.6710 

x 10 5 

0.42685 

0.16336 

x 10 18 

0.138255 

0.0042972 

426.85 

2.7374 

x 10 5 

to 

oo o 

CO rH 

10.333 

O 

^ 7 

CQ A, 
<?2 
^ X 

rH 

electronic 

charge 

0.62786 
x 10 19 

6.27968 
x 10 18 

3 = 

®2 

CQ 

C£> X 

CO 5 

CQ O 

co 

<£> X 

Tf 

■ f> 

oSs 
co 9—< 

^ X 

CQ 

2.62822 

x 10 19 

1.00587 

CO 

CQ 2 

rH O 
to rH 

2* 

U 

CO ** 

CQ X 

2.6288 

x 10' 8 

sa, 

00 r 

«> X 

rH 

2.6279 

X 10 19 

63.61979 

x 10 19 

- 

6.1572 

x 10 19 

liter atm 

05 7 

O 

°0 rH 

05 X 

n 

oo © 
*X 

o 

GO - 

© © 
® 2 

® X 

10.409 

CQ 3L 
tO © 
tO ^ 

co X 

0.041311 

00 « 
o 7 

S 2 

tO v 

rH *• 

0.013381 

4.1589 

x 10- 4 

41.3116 

26.5041 

X 10 3 

41.3066 

x 10 3 

rH 

15.7183 

x lO 22 

0.096782 

g cal 

0.238948 

0.238959 

2.3892 
x 10-* 

453.720508 

8.6004 
x 10 5 

1.000236 

0.382811 
x 10 19 

0.32397 

0.010069 

1000 

0.64171 

rH 

24.21212 

0.38057 

x 10 19 

2.3427 

hp hr 

to i 

CQ o 

c- -h 

CO x 

•— 1 w 

S © 

C- rH 

** X 
o 

1.3412 
x 10 10 

CQ ▼ 

S © 

CT5 

05 X 

o 

CO 

rH 

1.5593 
x 10- 8 

c- 8 

l> 

CO o 

05 h 

b X 

to *- 

§o 
o ^ 

lO X 

05 7 
co O 

to rH 

s* 

0.0015593 

- 

05 „ 

o 7 

05 O 
to rH 

2 x 

to 

S© 

co X 

5.9328 

x 10 28 

3.6530 

x 10-« 

g cal 
(mean) 

05 » 

So 

CO 

CQ X 

N x 

1.4333 
x 10 9 

0.25198 

860.585197 

n 

1 

O 

X 

rH 

3.82718 
x 10 23 

05 7 

So 

<n —i 
co X 

►-0I X 
899001 0 

rH 

CQ 

CO 

tO 2 

S X 

CO 

9.9987 

x 10* 4 

242.0624 

x 10 4 

3.8048 

X 10 23 

7 

CQ O 

S 

CO X 

CQ A 

ft poundal 

23.730 

41.44218 

St 

CO o 

t"~ rH 

^X 

CQ A 

O ,r 

CO O 

O rH 

X 

CQ * 

8.542335 
x 10 7 

99.334 

38.01712 
x 10-' 9 

32.174 

- 

99333.78 

63.72928 
x 10 8 

99.3219 

2404.5 

05 7 
*2 
CO x 

232.71 

ft lb 

0.73756 

0.73768 

7.3736 
x 10-" 

L6LLL 

2.6552 
x 10 8 

3.0874 

co 2 

rH 1 

00 o 

rH rH 

-< X 

- 

I801E00 

3087.4 

O » 
§2 

5 x 

3.0871 

74.735 

c- 2 

r- o 

rH rH 

^ X 

7.2330 

ev 

0.62419 
x 10 19 

0.62429 
x 10 19 

7.3756 

X 10 9 

§§, 
OO ^H 
to 

co X 

2.24691 
x 10 25 

2.612944 
x 10 19 

rH 

8.4628 
x 10 18 

CQ r- 

S 2 

CO ^ 

CQ X 

2612.85593 
X 10 19 

1.67324 
x 10 25 

2.61254 
x 10 19 

63.24792 
x 10 19 

0.99414 

6.12121 
x 10 19 

g cal 
(mean) 

0.23889 

0.23893 

2.388 
x 10 8 

251.98 

o° 

O rH 

CD 

- 

3.8272 

X 10 20 

0.32389 

O 

CO 

o 

o 

rH 

o 

o 

1000 

O „ 
co o 

rH r-H 

^ X 

CO * 

0.99987 

24.206 

3.8048 

X 10 20 

2.3427 

kwh 

GO 7 
£2 
<N X 

go 7 
£2 

<N X 

<$ 7 

-2 
M X 

o 

CO o 

05 rH 

(N X 

- 

1.163 
x 10 8 

4.451 

X 10" 28 

CQ 7 

c- ^ 
CO x 

t" o 

rH rH 

*+ X 

0.0011628 

| 

0.7457 

1.163 
x 10 8 

tO 7 

s ° 

GO rH 

ci x 

4.425 

X 10 28 

2.7235 
x 10“ 8 

Btu (mean) 

to ▼ 

S 2 

05 X 

9.4821 

X 10-“ 

9.4805 
x 10" 

- 

3413.0 

S896S00 0 

1.5188 
x 10 22 

0.0012854 

CQ 7 

tO ' 

05 2 
05 ^ 

co X 

3.9685 

2545.0 

0.003968 

0.09607 

§T 

rH O 

tO ^ 

X 

0.0092972 

erg 

o 

rH 

iOl x 
S9I000 I 

- 

1.0548 
x 10 10 

3.6000 
x 10' 3 

®2 

rH 

Tt X 

C- M 

o 7 

§2 

CO 

r-i X- 

CQ 

go 

« X 

rH 

4.21402 
x 10 5 

4.186 
x 10 10 

tO « 

S© 
oo Zl 

CO ^ 

CQ X 

4.1855 
x 10’ 

101.328 
x 10 7 

2 

CQ i 

05 O 
to *— 1 

X 

to 

CO ^ 

co 2 
o ^ 

°P X 

05 

joule 

(international) 

0.999835 

rH 

CO 
iO r- 
GO O 
05 rH 

8s x 
o' 

1054.625958 

3.599406 
x 10 8 

4.185313 

© 2 
2b 

O —' 

^ X 

rH 

1.355596 

0.042133 

4185.28838 

2.685156 
x 10 8 

4.184809 

101.311280 

1.592437 
x 10 19 

9.805031 

joule 

(absolute) 

rH 

S91000 1 

o 

rH 

X 

1054.8 

3.6000 

X 10 8 

CO 

CO 

1.60207 

X 10 19 

1.35582 

Tj* 

o 

o 

4186 

2.6845 

X 10 8 

4.1855 

101.328 

c- 2 

CQ i 

05 O 

to rH 

^ X 

9.80665 


II II II II II II II II II >1 II II II II II 


C 



£ 

CO 

is 


C 

s 

s 

<D 

c 


a 


p 



1 erg = 1 dyne cm 































876 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


ev 

ergs 

ft lb 

10*1 


I0 20 | 


IO' 3 


IO 3 ' 


IO' 9 


IO' 2 


I0 30 


10 18 


10" 


I0 29 


IO 17 


10“ 


I0 28 


10 16 


IO 9 


I0 27 


K)° 


IO 8 


io 26 


IO 14 


IO 7 


I0 25 

r 

10 13 


10 6 


IO 24 


10 12 

: 

10 9 

= 

IO 23 


10" 

- 

10 4 

r 

IO 22 

i 

10* 

“ 

10 J 

\ 

IO 21 

1 

IO 9 

“ 

10 2 


o 

M 

o 

=- 

IO 8 


10' 


IO 19 

r 

IO 7 


10° 


io 18 


IO 6 

— 

10 1 

— 

IO 17 

■ 1'. 

10 5 

— 

IO 2 

— 

IO' 6 

=- 

IO 4 

— 

IO 3 

— 

10 15 


10 ! 

r 

IO 4 

— 

10 14 

r 

10 2 

r 

IO 9 

— 

10 15 

_ 

10* 

r- 

10* 

— 

io 15 

; ” 

10° 

— 

IO- 7 



Fig. 22-5. Energy Conversions [3] 








PHYSICAL CONSTANTS AND CONVERSION FACTORS 


877 



Fig. 22-5 ( Continued ). Energy Conversions [3] 










Table 22-16. Spectroscopic Energy Conversions and Equivalences [5] 


878 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 



II II II II 



bJD 

u 

a> 


o 

> 

c 

o 

4- 

U 


c 

o 


o 

a> 


cn 

s- 

a> 

a 

a) 

o 


>> 


u 

a; 

X 

£ 

3 

C 

a> 

> 

03 

£ 



O 

o 


Fig. 22-6. Spectroscopic Energy Conversions 














Table 22-17. Power Conversions [2,4] 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 


879 


cd 

05 

be c/2 

£ 

o 


I 

be a; 

C/2 

£ 


a 

JS 


i 

cd 

$ 

CO 

be 

s- 

0) 


G 

s 

13 

cd 

be 


i 

G 

£ 

G 

cc 


I 

CD 

CD 

CO 

-O 




g 

G 

Sh 

0) 


£ 


CO 

-G 

G 



oo 


o 

o 



05 



£■ -r 

to 2 L 

CM O 
co ^ 

9 x 

*0 -r 

11 -r 

rH « 

i> 

CO 


a> o 

oo o 

co o 

CM O 

t> ' 

05 

05 


r-H 

rH 

CM r-H 

Tf r-H 

05 2 

CO 

05 


9 x 

o X 

9 x 

c- * 

2* 

O ^ 
r-H X 

o 

co 

05 

05 



rH 


rH 




05 


r-H 

00 

to 

to 

rH 

co 

Tf 

to 

9« 

■** 

N. 


CO 

05 *? 

05 

rH 

o 

CM 

o 

CM 

00 

CO 

co 

CM 

05 

CM 

■'t 

r-H 

<X> © 

rH rH 

o 

05 

CO 

o 

rH 

05 1. 

05 2 

rH 


rH 

t> 

rH 

• X 



9 x 
o 

o 

o 

d 

rH 


r-H ^ 

co 

t> 


0.0013410 

0.0013432 

0.023575 

0.0018182 

0.093557 

cm2 

rH 1 

Tf O 

CO 1-t 

H x 

rH 

0.01315 

to 7 
90 

9 *“ 1 

X 

o 

r- ^ 

to 

So 

o » 
oo © 

CM 

00 JG 
lO ° 

« 

CO o 


Ph. 05 

io o 

to 

co bL 
co 9 

to 

CO 2L 

CO 2 

X 

O 

9 x 

tO rH 

2x 

to ^ 

9 X 

r-H 

Sx 


Tt 1—1 

t> X 

o rH 
9 x 

o 1-1 

9 x 

rH 

rH 


rH 

CO 



05 

05 

00 

CO 

Tt* 

r-H 

to 

CO 

r-H 

CM 

05 

rH 

to 

co 

CO 

Tf 

05 

rH 

CO® 

go 

SrH 

00 

00 

co 

CO 

to 

o 

CO 

02 

o 

o 

CM 




d 

rH 


o 

d 

o 

o 

d 


H X 

rH 

o 

rH X 

CD 

o 


Tt 




co 

05 

t> 

to 


05 

00 

CO 

tO 

05 

05 

CO 

UO 

rH 

CM 

rH 

t- 

c- 

05 

CO 

9 

® 
oo i 
oo o 

CO ,-H 

oo 

r-H 

9 

co ® 

s® 

to ^ 

o 

o 


o 

co 

to X 

03 

Tf 

to 

td x 

d 

o 


o 




o 

CO 

tO 

t> 

co 

oo 

t> 

00 

CO 

co 

co 

05 

rH 

t> 

to 

9 

co ® 

“s 

co ^ 

o 

to 

05 

05 

CM 

CO 

05 u; 

05 l 

CM o 

CO rH 


o 

CM 


rH 

to 

CM 

CM w 

o 

d 

rH 


to 

t> x 



^ x 

lO 


co 


00 

to 

05 

CO 

cn 

CO 


05 



97 

co 

o 

i-r 

00 

05 

05 

05 

rH 

to 

to 

to 

9 

to 

rH 

to 

to 

t> 

05 

So 

05 ~ 

05 ^ 

9 x 

o 

to 

id 

to 

o 

co 

05 

UO o 

O r-H 

co 

d x 

o 


rH 


CO 

o 



uo 

CO 

r-H 

o 

00 

CM 

00 

to 

co 

r>. 

1 

o 

o 

t> 

to 

CO 

co 

S'? 

co o 

r-H 

o 

to 

to 

I> 


id 

o 

O* rH 


o 


CO 

05 

X 


CO 

00 V 


rH 

rH 

rH 

co 

rH 


05 

05 X 




G 


S-4 

05 



cG 



G 

0) 


a 

cr 


G 

o 


£ 

£ 2 
-*-> G 

T3 

G 

G CT 

a> i 

1 

CD 

o 

CO 

a 

r! 

CD 

0) 

C/2 

£ 

G 

r-* 

c £ 

G C 

G 

O 

CD 

£ e 

^99 • i-h 

C 

be 

s- 

05 

» 

• 

U 

K 


H 

_o 

_ G 

0) 

Cfl 

0) c 

9 

>H 


hh 

G 

r- 1 PQ 

(h 

j- 1 d 
o ® 

G3 

D 

05 

HH 

-*-> 

G 

E S 

a> 

-G CD 

G 


05 

G 

'o 

CO 

C 

tH 

<D 

aJ 2 
2 | 

cu^ 

T3 7 

G c-> 

S w> 

£ ^ 

O 

CD 

05 

C/2 

Sh 

05 

£ 

£ 7 
£ ^ 

-Q 

G 

■ 

C 

| 5 

G 

-G •- 
.22 £ 

G 5 

£ d 
be 2 

Sh 

8. 

05 

Cfl 

(h 

o 

2 efi 

G _ 

be £ 

• * 

H> 

03 

d t- 
fc- <D 

O ttJ 

-2 .£ 

• H H 

be 

s_ 

-2 bo 
•- 1 -X 

£ 


CQ CD 

2 - 

c 

05 

-C 

-X ^ 

rH 

^H 

rH 

rH 

rH 

rH 

rH 

rH 


"C 

G 

O 

CD 

05 

CO 

S-. 

0) 

a 

S- 

05 

*-> 

CD 


C 

0) 

cd 

£ 

G 

s- 

bc 


CD 

05 

CO 

£ 

C5 

be 





















Table 22-18. Electrical Unit Conversions [4-6] 


880 


PHYSICAL CONSTANTS AND CONVERSION FACTORS 



















PHYSICAL CONSTANTS AND CONVERSION FACTORS 


881 


Table 22-19. Prefixes 


Multiples and 
Submultiples 

Prefixes 

Symbols 

10 

deca 

da 

10 2 

hecto 

h 

10 3 

kilo 

k 

10 4 

10 kilo 


10 5 

100 kilo 


10 6 

mega 

M 

10 7 

10 mega 


10 8 

100 mega 


10 9 

giga 

G 

10 10 

10 giga 


10 n 

100 giga 


10 12 

tera 

T 

10 13 

10 tera 


10 14 

100 tera 


io- 1 

deci 

d 

10- 2 

centi 

c 

IO 3 

milli 

m 

IO 4 

100 micro 


IO 5 

10 micro 


IO" 6 

micro 

P- 

10 7 

100 nano 


io-« 

10 nano 


IO 9 

nano 

n 

10-jo 

100 pico 


IO 11 

10 pico 


IO 12 

pico 

P 




















































































































Index 


Abbe prism, 308 

Abbe sine condition, 282, 428 

Abbe V number, 403, 409 

Aberrations 

Abbe V number, 403, 409 
afocal systems, 406 
angular, 381 

astigmatism, 384, 389, 400, 439, 448 
axial, 384, 389, 400, 404 
Bouwers system, 446 
catadioptric system, 448 
chromatic variations, 386 
correction, 385 
lateral color, 385, 390 
longitudinal, 384, 389, 400, 404 
off-axis, 385, 390 
refracting lens, 436 
stop shift theory, 404 
transverse, 385, 390 
coma 

calculation, 400 
catadioptric system, 448 
description, 382 
ray tracing, 388 
rim ray curve, 387 
spherical reflector, 438, 439 
stop shift theory, 404 
transfer function, 629, 634 
correction, 385, 406, 570 
distortion 

calculation, 400 
description, 389 
shock wave, 842 
stop shift theory, 404 
transfer function, 630 
window induced, 826 
field curvature. 384 
higher orders, 399 
image curvature, 630 
longitudinal, 381, 386 
Petzval surface, 384, 400, 439, 440 
plane parallel plate, 447 
plots, 399 
residual, 386, 407 
Seidel, 382, 630 
spherical 
blur spot, 435 
Bouwers system, 446 
calculations, 400 
catadioptric system, 447 
description, 382, 388 


fifth-order, 633 
longitudinal, 386, 399 
resolution limits, 414 
rim ray curve, 387 
refracting lens, 436 
spherical reflector, 438 
transfer function, 629, 631 
transverse, 399 
stop shift theory 7 , 404 
symmetrical principle, 406 
third-order, 399, 402, 404 
transverse, 381, 399 

variations with aperture and image size, 385 
wave-front, 616. 630 
wave theory, 630 
zonal, 386 

Aberration-free systems, 629 
Abney grating mounting, 312 
Absolute stability, see Stability, absolute 
Absorptance 

approximations to band models 
nonoverlapping case, 201 
strong-line, 200 
table, 199 
weak-line, 197 
definition, 784 
directional, 26 
Doppler coefficient, 196 
Doppler line shape. 190 
Doppler broadening, 191 
Elsasser model 

compared to statistical model, 195 
development, 192 
strong-line approximation. 200 
weak-line approximation, 198 
weak-line model, 197 
Lorentz broadening, 191 
Lorentz coefficient, 196 
Mayer-Goody model 

approximations of band absorptance, 199 
development, 194 
models, table, 199 
quasirandom model 
development, 196 
strong-line approximation, 201 
weak-line approximation, 199 
random Elsasser model 
development, 196 
strong-line approximation, 201 
weak-line approximation, 199 
single spectral line with Lorentz shape. 190 


883 


884 


INDEX 


solar 

ultraviolet, 792 
various materials, 800 
square coefficient, 196 
square root region, 191 
statistical model 

approximation of band absorption, 199 
comparison with Elsasser model, 195 
strong-line approximation, 191 
total, 28 
Absorption 
See also Absorptance 
atmospheric bands, 238 
nonequilibrium conditions, 28 
selective filters, 286 
wavelengths of molecular groups, 325 
Absorption coefficient, 189, 190, 358 
Absorptivity 
definition, 784 

measurement techniques, 796 
ratio to emissivity, 787, 791, 804 
selected materials, 805 
Accelerated life test, detectors, 461 
Acceleration 
conversion factors, 870 
definition, 856 
Acetylene smoke, 356 
Achromats, 282, 409 
Acquisition, 744-748 

Admiralty Research Laboratory blackbody slide 
rule, 17 

Aerial targets, see Targets 
Aerodynamic conditions, 827 
Aerodynamic flow, optical influences, 826 
Aerodynamic heat transfer, 827, 838 
Aerographic film, 571, 576 
Aerosols, 141, 208 
Afocal system, 406, 422 
Afterburners, 60 
Air, 839, 851 
See also Atmosphere 
Air mass, 97 
Air-to-air intercept, 744 
Airglow, 104 
Airy disc, 410, 440 
Airy pattern, 616 
Airy system, 623 
Albedo 

cylindrical geometry, 135 
definition, 785, 812 
hemispheric geometry, 814 
planar geometry, 816 
spectral distribution, 790 
Alumina, 359, 804 
Aluminum 
a/e ratio, 804 
emissivity, 359 
reflectance, 351 
Aluminum foil, 77 
Aluminum paint, 78, 809 
Aluminum-oxygen group, 325 
Alzac, 354 

AM-FM discriminator, 745 


Amici prism, 308 
Amplifiers 

See also Preamplifiers 
feedback, 686 
grounded cathode, 595 
high-frequency response, 595 
low-noise, 597 
Miller effect, 595 
noise sources, 584 
power supply, 597 
transistors 
bias point, 598 
bias stabilization, 598 
circuits, 602 
field-effect, 606 
junction, shot noise, 586 
low-noise, 606 
noise, 597, 599, 602 
planar, 608 
silicon, 601 
stability, 598 
transformer coupling, 604 
vacuum tube, 592 
very high impedance, 596 
Amplitude, complex, 615 
Anamorphic systems, 424 
Angle 

conversion factors, 865 
definition, 855 
Angular aberrations, 381 
Angular dependence of reflectivity, 793 
Anode current fluctuations, 585 
Antimony, 805 

Apertures, see Detectors; Filters; Optics 
Aplanatic lenses, 380, 427, 434 
Aplanatic surfaces, 427 
Apodization, 636 

Luneberg theorem, 621 
Area, 855 
Argon, 179 

Arsenic-modified selenium glass, 321 
Arsenic trisulfide, 282, 294 
Artificial sources, 32 
Asphalt, 83, 89 
Aspheric surfaces, 385, 402 
Astigmatism, 384-389, 400, 439, 448 
Astronomical data, 794 
Astronomical telescope, 422 
Atmosphere 

See also Backgrounds, sky; Stratosphere 
absorption, 178, 189, 237, 266 
carbon dioxide, 180, 238 
carbon monoxide, 249 
composition, 179 
density, 177 

Lorentz broadening, 178, 189 

methane, 250 

nitrogen, 178 

nitrous oxide, 246 

oxygen, 178 

ozone, 185 

particles, 187 


INDEX 


885 


pressure, 189 
radiation, 97, 737 
scattering, 203 
scintillation, 210 

See also Stars, stellar scintillation 
solar spectrum measurements, 227 
spatial filtering, 737 
temperature, 96, 177 
transmission, 252, 737 
water vapor, 181, 244 
Aurora, 101 

Autocorrelation function, 649, 726 

Azimuth coverage, 738 

Axial chromatic aberration, 384, 404 

Axial cone, 377 

Axis of symmetry, 844 

Back focus, 375 
Backgrounds 
See also Terrain 
discrimination, 732, 744 
earth, 115 
fluctuations, 730 
marine, 166 

radiation, 737-739, 752, 826 
radiation-noise-limited system, 752 
sky, 96, 143 
See also Clouds 
stellar radiation, 107 
temperature, 462 

Background-limited D*-Q trade-off, 753 
Baffles, 379, 442 
Baird-Atomic filters, 299 
Bands, see Spectral bands and lines 
Bandpass interference filters, 290 
Bandwidth, spectral, 763 
Bang-bang systems, 715 
Barium fluoride, 294 

Barnes Engineering Co. catadioptric systems, 
285, 286 

Barnes Engineering Co. nitrogen pressure gen¬ 
erator, 530 

Barr and Stroud glass, 321 

Base width (BW) filters, 287 

Basic period filters, 290 

Bausch and Lomb filters, 299 

Bausch and Lomb glass, 321 

Beryllium, 805 

Bessel function, 648 

Beutler radius grating mounting, 312 

Bias, 502 

Bias point, transistors, 598 
Bilayer, fictitious, 292 
Bilinear transformation analysis, 702 
Birefringent crystals, 298 
Blacks, 359-364 
Black enamel, emissivity, 359 
Blackbodies 
cavity sources, 51 
curves, 17 

detectivity, 466, 467 
emittance, detectors, 501 
gold point, 44 


Kirchhoff’s law, 9 
NBS standards, 38 

noise equivalent power, detectors, 466 

Planck’s function, 795 

quantum rates, 10 

radiance, 96 

reference, 761 

responsivity, detectors, 464 

simulator, 32 

slide rules, 11 

tables, 21 

2400°K, 47 

vertical, 45 

Blackened chopper, 760 
Blazed diffraction gratings, 310 
Block blackbody slide rule, 11 
Blocking filters, 291 
Blur spot, 435, 452 
Bode diagrams, 667 

Bode method of linear systems analysis, 668, 678 
Boil, 210 

Bolometric detectors, 459, 497, 498 
Boron nitride, emissivity, 359 
Boundary-layer flow, 170, 828, 831, 846, 848 
Bouwers-Maksutov optical system, 285, 445 
Boxcar circuits, 696 
Brick, daytime radiance, 146 
Broadband transformer, 604 

Cadmium, 805 

Cadmium-germanium detectors, 491 
Calcium aluminate glass, 317, 321 
Calcium fluoride 
prisms, 309 

reflection coefficient, 298 
substrate, 294 

Canonic form, differential equations, 704 
Caps, auroral, 101 
Capacitance impact effects, 592 
Carbon arc source, 49 

Carbon-chlorine group, absorption wavelengths, 
325 

Carbon dioxide 

atmospheric absorption bands, 238, 239, 240, 
242, 243 

atmospheric composition, 179, 180, 181 
exchange cycle, 180 
Carbon-hydrogen group, absorption 
wavelengths, 325 

Carbon-oxygen group, absorption wavelength, 
325 

Carbon filaments, 38 
Carbon monoxide 

atmospheric absorption, 249 
atmospheric distribution, 179, 187 
Carbonyl group, absorption wavelength, 325 
Cascaded optical systems, 626 
Cascaded thermoelectric cooling systems, 525 
Cascode connection, preamplifiers, 608 
Cassegrain radiometers, 762 
Cassegrain telescope, 442 
Catadioptric systems, 285, 443, 624 
Cavities, 32, 51 


886 


INDEX 


Celestial sphere, 785 
Cell bias, detectors, 502 
Cesium bromide prisms, 297, 309 
Cesium dioxide film, 295 
Cesium fluoride film, 295 
Cesium iodide, prisms, 309 
Childs’s law, 585 
Chiolite film, 295 
Choppers, 759, 760 
Chopping system, 759 
Christiansen filters, 295 
Christiansen wavelengths, 296 
Chromatic aberration, 384-390, 400, 404, 436, 
446-448 
Chromium, 805 

Circular aperture filters, 654, 655 
Circular apertures, 616-618, 624, 630 
Circular scan, 736 
Circular-sectored reticles, 653, 655 
Cirrus clouds, 99, 122 
Clamping circuits, 696 
Claude cycle analysis program, 545 
Clock plot, aberrations, 388 
Closed-loop describing function, 708 
Closed-loop transfer function, 722 
Clouds 

attitudes, 118 
cirrus, 99, 122 
cover, 119, 124, 737 
infrared transmission, 258 
meteorology, 118 
nacreous, 124 
noctilucent, 124 
stratospheric, 124 
top, 122 

Coal tar pitch, reflectance of, 84 

Coarse grain film, 572 

Cobalt, 805 

Codit silver paint, 90 

Coddington’s equations, 392-395 

Coherent illumination, 626 

Cold junction, heat balance at, 550 

Cold trap, 533 

Color reversal film, 571 

Colored glass filters, 306 

Columbium, 805 

Collimator, 440 

Coma, 400-404, 629-634, 382-389 
Commutative condition, 614 
Compensation 
high-frequency, 590 
image motion, 755 
singular points, 723 
Complex amplitude, 615 
Computers, 394, 640 
Concave diffraction gratings, 310-311 
Concrete, 88-92, 149-156 
Condensers, optical, 424-430 
Conductance, 786 
Conduction 
definition, 785 
thermal joint, 786 

Conductivity, thermal, of optical materials, 331 


Cone, aperture, 377 
Cone condensers, 427, 430 
Conical lenses, 843 
Conjugate functions, 647 
Conrady G sums, 341, 403 
Constant deviation prism, 308 
Contact conductance, 786 
Contours 
s-plane, 673 
sensitivity, 510 
Contrast ratio, 623 
Control systems 
asymptotic stability, 704 
block diagram, 667 
design, 718, 727 
feedback, 662 
integral control, 721 
linear systems, 662 
nonlinear systems, 703 
ramp inputs, 672, 687, 690, 694 
sampled-data system, 695 
Conversion factors 
acceleration, 870 
angles, 865 
density, 867 
electrical units, 880 
energy, 855, 858, 875 
force, 871 
length, 860 
mass, 866 
power, 879 
pressure, 872 
ranges, 863 

spectroscopic energy, 878 

temperature, 873 

time, 868 

torque, 871 

velocity, 869 

volume, 864 

work, 875 

Convolutional integral, 621, 648 
Coolers, See Detectors, cooling systems 
Copper 

Cu-Cu 2 0 detectors, 472 
Ge:Cu detectors, 490 
reflectance, 351 

thermal radiation properties, 805 
Corner cube mirror, 773 
Coming glass, 317, 318, 322 
Correlation quality, 638 
Corrugated metal, 93 
Cotton, reflectance of, 76 
Coupling, transistor-transformer, 604 
Cross-correlation function, 726 
Crossover frequency, 215, 664 
Cryogenic data, 521 
Cryolite, 295 

Cryostat coolers, 523, 532 
Crystals, birefringent, 298 
Cube mirror, comer, 472 
Curvature 
field, 384-389, 404 
image, 630 


INDEX 


887 


Petzval, 385, 439-440 
Cutoff slope, filter, 287 
Cutoff wavelength, filter, 287, 468 
Cuton slope, filter, 287 
Cuton wavelength, filter, 287 

D*, see Detectivity 

D*-Q tradeoff, 753 

D**, see Detectivity 

Data enumeration, 460 

Data function, equivalent continuous, 697 

Day airglow, 106 

Decimal equivalents of common fractions, 862 
Decoy discrimination, 744 
Deep space probes, 811 
Defect, fidelity, 637 
Definition, optics, 410 
Defocusing coefficient, 387, 618, 630, 632 
Degrees, radian equivalents, 865 
Delta function, Dirac, 648, 662 
Demodulation process, 745 
Density 
definition, 856 
conversion factors, 867 
flux, 730 

photographic film, 573 
Depth of field, 380 
Depth of focus, 380 
Describing function, 706-708 
Detection 
definition, 738 
probability, 738, 740 
range, 739 
synchronous, 610 
Detective quantum efficiency, 462 
Detectivity 
blackbody, 466 
cutoff wavelength, 468 
blackbody D*, 467 
detective quantum efficiency, 468 
maximized D*, 467 
peak wavelength, 467 
photon-noise derivation, 513 
spectral, 466 
spectral D*, 467 
spectral D**, 467 
system design considerations, 515 
theoretical limit, 512 
Detector-noise-limited system, 751 
Detectors 

See also Detectivity 
accelerated life test, 461 
acceleration specifications, 461 
apertures, 459 
array, 755 
cooling systems 
design criteria, 522 
dewars, 559 
direct contact, 523, 526 
expansion engine cooling systems, 523, 540- 
545 

Joule-Thomson (cryostat), 523, 532 
limitations of fluid cooling, 520 


temperature ranges, 525 
thermoelectric (Peltier) coolers, 523, 546 
cruciform, 747 
Cu-Cu 2 0, 472 
data enumeration, 460 
dewar flask, 459 
front-end description, 747 
GaAs, 471 

Ge:AuSb (rc-type), 488 

Ge:AuSb (p-type), 489 

Ge:Cd, 491 

Ge:Cu, 490 

Ge:Hg, 492 

Ge-Si:Au, 496 

Ge-Si:Zn, 495 

Ge:Zn, 493 

Golay, 772 

HgTe, 494 

InAs, 481, 482 

InSb, 484-487 

InSb bolometers, 497 

PbS, 474-476 

PbSe, 477-479 

PbTe, 480, 836 

noise, 464, 588 

parameters, 459, 462 

photodetectors, 458 

radiation transducer, 458 

saturation, 833 

Si, 473 

symbols and units, 459 
system, multiple, 753 
Te, 483 

test procedures 

frequency response, 504 
NEP, determination of, 501 
noise spectrum, 510 
optimum bias, 502 
pulse response, 505 
sensitivity contours, 510 
spectral response, 508 
spinning mirror technique, 506 
time constant, 504 
thermal 

Golay cells, 458, 459, 500 
thermistors (bolometric), 458, 497, 498 
thermocouples (thermovoltaic), 458, 459, 
499 

thermopiles (thermovoltaic), 458, 459 
time constant, 504 
vacuum environment, 461 
windows, 359 
Deuterium, 520 
Deviating prisms, 308 
Dew point, 184 
Dewars, 459, 557 

Dielectric constants, optical materials, 329 
Differential equation, canonic form, 704 
Diffraction, See Optics, diffraction theory 
Diffraction integral, Kirchhoff, 635 
Diffraction gratings, 309-312, 763-772 
Diffraction limitation of optical system, 440 
Diffraction-limited systems, 448 


888 


INDEX 


Diffraction pattern, 411, 635 
Dimensionless wave number, 288 
Diode, space-change-limited, 586 
Dirac delta function, 648 
Direct-contact cooling systems, 522, 526 
Direct-ratio spectrophotometer, 776 
Directional absorptance, 26 
Directional emissivity, 23, 26 
Directional emittance, 785 
Directional reflectance, 23, 26 
Directional reflectivity, 88-93, 793 
Discrimination, 732, 737 
decoys, 744 

Discriminator, AM and FM, 745 
Disc, Airy, 410, 440 
Dispersing elements, 763 
Dispersing prisms, 309 
Dispersion 

angular, of thin lenses, 340, 404 
Herzberger equation, 337 
optical materials, 328 
equation, 335 
power, 339 
refractive index, 338 
partial, 403 
reciprocal, 403 

Displacement error, systems, 663 
Displacement input, linear systems, 689 
Displacement law, Wien, 10 
Displacer piston expander, 544, 545 
Distance measures, 863 
Distortion, 384-389, 400-404, 630, 826, 842 
Distribution 
albedo radiation, 790 
atmospheric constituents, 181 
illuminance, 624 
innerloop gain, 725 
Poisson, 585 
spectral, 761 
stellar radiation, 110 
Diurnal variations 

atmospheric scintillation, 223, 226 
radiance of concrete, 149 
radiance of grass, 143 
radiance of backgrounds, 144 
Domain 
frequency, 637 
spatial, 623 
of linear systems, 614 
spatial frequency, 628 
Doppler coefficient, 196 

Doppler effect on Lorentz broadening of spectral 
lines, 189 

Doppler half-width, 190 
Doppler line shape, 190 
Double monochromators, 765 
Double-beam optical null spectrophotometer, 
775 

Double-beam photometer, 772 
Double-pass spectrometer, 771 
Double-pass system, monochromator, 773 
Dove prism, 308 
Drag penalty, 827 


Drift, radiometer, 759 

Droplets, atmospheric, 188 

Dupont Flat Black paint, emittance, 356 

Dwell time, 730, 734, 739, 752 

Dyed-plastic filters, 307 

Dynamic transconductance, 586 

Dynasil glass, 318 

Eagle grating mounting, 312 
Earth 

as a background, 115 
astronomical data, 794 
emissive power curve, 790 
shine, 785 

spectral emissive power curve, 790 
Eastman Kodak Co. 
film and plates, 571-580 
filters, 299, 575, 580 
lenses, 282 
NOD-18 black, 359 
Ebanol, 805 

Ebert-Fastie plane grating mountings, 311 
Ecliptic, 785 

Effective optical thickness, 294 
Effective power, radiometers, 763 
Ektachrome, Aero film, 571 
Eikonal (wave-front aberration), 616 
Electrical units 

conversion factors, 880 
definition, 857 

Elevation angle, ocean spectral radiance, 169 
Elsasser model, 192-200 
Emissance, See Emissivity 
Emission 

atmospheric constituents, 96 
exhaust gases, 67 
ozone, 96 
sky, 96 
targets, 70 
Emissivity 
alumina, 804 
cavities, 32 
chopper mirror, 760 
definition, 784 
DeVos method, 33 
directional, 26 
Globars, 359 
Gouffe method, 32 
hemispheric, 793 
Kirchhoff’s law, 9 
materials, 359, 800, 805 
measurement techniques, 796 
NBS comparison standards, 44 
nonequilibrium condition, 28 
normal, 795 

partially transparent bodies, 23, 349 
radiant, 74 

ratio to absorptivity, 787, 791, 804 

Sparrow method, 37 

terrain, 75, 142 

total, 28 

water, 167 

windows, 827, 835 


INDEX 


889 


Emittance 
blacks, 362, 356 
definition, 784 
radiant, 23 
total, 785 

Emulsion, infrared, 570 
Enamel 
black, 359 
Chinese red, 80 
white, 81, 359 

Endless checkerboard, 651, 654-655 
Energy 

conservation of, 27 
conversion factors, 855, 858, 875 
definition, 856 
distribution, 410 
Fermi, 544 
Engelhard glass, 318 
Entrance aperture, flux density, 730 
Entrance pupil, 377, 379, 733 
Entrance slit, monochromator, 766 
Episcotister, 653 

Epoxy-type molding compound, 357 

Equatorial inclination, 785 

Equilibrium constant, 66 

Equivalent, Thevenin, 590 

Equivalent continuous data functions, 697 

Equivalent layer, Herpin, 293 

Equivalent power detectors, 466 

Equivalent noise voltage, 607 

Equivalents of common fractions, decimal, 862 

Erectors (optical systems), 424 

Errors 

acceleration, linear systems, 663 
control methods, 721 
displacement, linear system, 663 
generation, tracking systems, 750 
modulation techniques, 744 
rate, linear systems, 663 
Error plus error-rate system, 714 
Ester group, absorption wavelength, 325 
Eurasian land mass, 122 
Exhausts, 59-67 
Exit heating, rockets, 68 
Exit pupils, 377 

Expansion, thermal, of optical materials, 332 
Expansion-engine cooling systems 
characteristics, 523 
commercially available, 541 
displacer piston expander, 544, 545 
Gifford-McMahon piston expander, 540 
turbine expanders, 544, 545 
Exposure, films and plates, 573 
Exposure meters, 570 
Extinction coefficient, 351 

f number, 379, 429 

Fabrics, reflectance, 76 

Fabry-Perot interferometer, 290, 778-781 

Farrand Optical Co. filters, 299 

Feedback, negative, 593 

Feedback amplifiers, 686 


Feedback control system, 662 
Fermi energy, 544 
FET preamplifier design, 608 
Fidelity defect, 637 
Field, depth of, 380 
Field curvature, 384-389, 404 
Field-effect transistors, 606 
Field lens, 424-429 
Field of flow refraction, 839 
Field of view, instantaneous, mapping system, 
734, 739, 745 
Field stop, 378 

Fifth-order aberrations, 382, 633 
Figure-of-merit, cooling systems, 548 
Films and plates 
aberration correction, 570 
aerographic, 571, 576 
characteristic curves, 573 
coarse grain, 572 
color reversal, 571 
density, 573 

Ektachrome Aerofilm, 571 
exposure, 573 

extremely high-contrast, 572 
H and D curves, 573 
high contrast, 571 
high-speed, 571, 575, 579, 580 
infrared, 571-580 
medium contrast, 572 
reciprocity characteristics, 574 
relative visibility curve, 573 
sensitometric characteristics, 573 
spectral sensitivity, 574 
spectroscopic, 572-579 
Filtering, spatial, 646, 737 
Filters, 

absorption, 306 
absentee layers, 288 
angle shift, 290 
background region, 287 
birefringent, 298 
blocking, 291 

characteristic admittance, 293 
Christiansen, 295 
circular aperture, 654-655 
circular-sectioned reticles, 655 
commercially available, 299 
dyed-plastic, 307 
film materials, 295 
fixed-field, moving reticle, 650 
Fourier transforms, 654 
infinite checkerboard reticle, 654-655 
infinite parallel-spoke reticle, 654 
interference 

analogies with transmission-line theory, 
293 

angle of incidence, 293, 350 
bandpass, 290 
characteristic matrix, 291 
description, 236 
effective optical thickness, 294 
fictitious bilayer, 292 
films, 294 


890 


INDEX 


filter matrix, 291 
Herpin equivalent layer, 293 
long-wavepass, 290 
narrowband, 291 
polarization, 298 
short-wavepass, 290 
spike, 292 

square-band, 291-292 
substrates, 294 
terminology, 286, 288 
line admittance, 293 
moving-reticle fixed field, 650 
moving-reticle scanning-field, 652 
narrow, radiometer use, 763 
plastic, 306 

rectangular aperture, 654-655 
scanning-aperture space, 650 
scanning-field, moving reticle, 652 
selective reflection, 297 
selective refraction, 298 
sintered metal, 533 
spatio-temporal output, 649 
spectral, monochromators, 766 
substrate materials, 294 
terminology, 286 
transmittances, 574, 580 
First-derivative control methods, 721 
First-order aberrations, 382 
First-order optics, 371 
First-order high-reflectance zone, 288 
Fish Schurman Corp. filters, 299 
Fixed field, moving-re tide space filters, 650 
Flame-temperature spectrometer, 772 
Flex-o-lite beaded paint, 92 
Flicker current noise, 586 

Flow, boundary-layer, see Laminar flow; Sep¬ 
arated flow; Turbulent flow 
Fluid cooling systems, 520 
Fluorite, 39 
Flux, meteoric, 793 
Flux density, 730 
FM-AM discrimination, 745 
Focal length, effect of shock wave, 844 
Focal plane, paraxial, 386 
Focal point, 371 
Focus 
back, 375 
depth of, 380 
paraxial, 634 
plane of best, 638 
sagittal, 384 
Fog, 188, 207, 211 
Folded optical systems, 442 
Force 

conversion factors, 871 
definition, 856 
Fore prism, 766 
Folded reflector, 441 
Four-element achromat, 284 
Fourier approach (cascaded system), 627 
Fourier transforms, 614, 621-622, 646-652 
Fractions, decimal equivalents, 862 
Fraunhofer lines, 227 


Fraunhofer plane, 616 
Fraunhofer receiving plane, 619 
Free filter range, 287, 290 
Frequency 
crossover, 215, 664 
spatial, 626 

Frequency domain, 637 

Frequency filtering, 737 

Frequency response, see Response, frequency 

Frequency shifting, space, 646 

Front-end description, 747 

Frost point, 184 

Fuels, high-energy solid, 67 

Fused silicate glass, 317 

G sums, Conrady, 341, 403 
Gain 

adjustment design methods, 718 
distribution, inner-loop, 725 
margin, 664 
system, 676 

Galactic concentration of stars, 109 
Galilean telescope, 422 
Gallium arsenide 
detector, 471 
reflection coefficient, 297 
transmission, 324 
Gallium antimonide 
reflection coefficient, 297 
transmission, 324 

Gallium phosphide, transmission, 324 
Gas-supply cooling systems, 532 
Gases, liquefied, 520 
Gaussian equation, 372 
Gaussian optics, 371 
Gaussian probability, 740 
Gaussian quadrature method, 631, 633 
General Electric Co. 
blackbody slide rule, 11 
glass, 318, 321 

Generation-recombination noise, 470, 587 
Geometry 

cylindrical, of albedo, 815 
hemispheric, albedo, 814 
incident beam, 25 
planar, albedo, 816 
planetary thermal radiation, 817 
problem, in systems design, 737 
projected solid angle, 24 
radiation, 22 
scattering angle, 117 
sea-surface, 168 
Germanate glass, 321 
Germanium 
filter film, 295 
filter use, 294 
detectors 

Ge:AuSb (n-type), 488 
Ge:AuSb (p-type), 489 
Ge:Cd, 491 
Ge:Cu, 490 
Ge:Hg, 491 
Ge-Si:Au, 496 


INDEX 


891 


Ge-Si:Zn, 495 
Ge:Zn, 493 
lenses, 282 

semiconductor properties, 324 
Germanium monoxide, 352 
Germanium-oxygen group, 325 
Gifford-McMahon piston expander, 540 
Gladstone-Dale constant, 839 
Glare stops, 379 
Glass 

arsenic trisulfide, 294 
calcium aluminate 
index of refraction, 323 
description, 317 
transmission, 323 
types, 321 
colored filters, 306 
Coming, No. 186SJ, 317, 318, 322 
filter use, 294 
fused silicate, 317 
fused quartz, 317 
General Electric Co., 321 
germanate, 321, 322 
high-silica-description, 317, 318 
lead silicate, 318 
lenses, 282 
NBS F998, 321 
nonoxide, 321 
silicate, 321 
suppliers, 318 

thermal radiation properties, 805 
Vycor, 317 
Global stability, 704 
Globar source, 48, 359 
Golay cells, 458, 500 
Golay detectors, 772 
Gold 

Ge:AuSb (n-type) detectors, 488 
GerAuSb (p-type) detectors, 489 
Ge-Si:Au detectors, 496 
reflectance, 79, 351, 805 
Gold black, 359 
Gold paint, 79 
Gold-point blackbodies, 44 
Gouffe method of calculating emissivity, 32 
Gradient methods of optical design, 409 
Graphite, 806 

Grass, daytime radiance, 146, 164, 157 
Grating spectrometers, 766 
Gratings, 309-312, 763-772 
concave mountings, 311 
Grazing incidence, 312 
Great circle arc, 115 
Gregorian telescope, 442 
Grey bodies, 70 

Grounded-cathode amplifiers, 595 
Grounding techniques, 611 

H&D curves, 573 
H&L layers, 288 
Half-width 
filters, 287 
spectral lines, 189 


Hardness, optical materials, 333 
Heat balance at cold junction, 550 
Hazes, 188, 207, 211 
Heat 

See also Temperature; headings beginning 
Thermal 
flow, 786 
flux, solar, 811 
load, of cooling systems, 521 
pumping, 550 
spacecraft, 68 
specific, 333 
transfer 

aerodynamic, 827 
flat plate, 828 
to hemisphere, 829 
radiant, 787 
radiation, 828 
separated flow, 838 
wall temperature, 829 
transfer coefficient 
definition, 827 
flat plate, 828 
wakes, 839 

Heat-energy balance, 73 
Helium, 179, 520 

Helmholtz’s reciprocity theorem, 24, 27 

Hemispherical emissivity, 793 

Hemispherical emittance, 785 

Hemispherical heat transfer, 829 

Hemispherical immersion lens, 433 

Hemispherical integration, 25 

Hemispherical shock-wave, 846 

Heraeus glass, 318 

Herpin equivalent, 290, 293 

Herzberger dispersion equation, 337 

High-contrast film, 571 

High-frequency compensation, 590 

High-frequency response amplifier, 595 

High-intensity sources, 49 

High-order aberrations, 399 

High-silica glass, 318 

High source impedance, 590 

High-speed flight, 826 

High-speed infrared film, 571, 575, 579, 580 

High-reflectance zone filters, 288, 290 

Holding circuits, 696 

Homogeneous paths, 263 

Horizon, 143, 737 

Horizontal paths, 252, 263, 266 

Humidity, 97 

detector requirements, 461 
Hydrogen, atmospheric, 179 
Hydrogen deuteride, 520 
Hydroxyl group, 325 
Hypersensitizing, 570 
Hypocycloid scan, 736 

Ice crystals, 122 

Ideal generator, noise power, 588 
Illuminance 
definition, 615 
distribution, 624 


892 


INDEX 


ratio, 615 

Illumination, coherent, 626 
Image 
blur, 386 
spot size, 449 
boil, 212 
contrast, 643 
curvature, 630 
dancing, 849 

enlargement (atmospheric scintillation) 213 

motion compensation, 755 

nutation, 747 

size aberrations, 385 

spot size, 410 

Immersion lenses, 427, 433-4 
Impedance 

amplifier, very high, 596 
detectors, 462 
high source, 590 
matching, 293 

matching circuit, detectors, 502 
open-circuit, filters, 293 
short-circuit, filters, 293 
source, 609 
vs noise, 598 
Impulse response, 614 
In-line spectrometer, 774 
Incidence, angle of, 293, 350 
Incident beam geometry, 25 
Incident radiance, 27 
Incident radiation 
albedo, 813 
planetary, 819 
solar, 813 
total, space, 821 
Inclination 
equatorial, 785 
planetary, 785 
Incoherent light, 621 
Inconel, 359, 806 
Inconel-X, 800 
Index of refraction 
air, 839 

boundary-layer effects, 847 
calcium aluminate glass, 323 
Coming No. 9752 glass, 322 
dispersion values of optical materials, 338 
NBS F998 glass, 321 
optical materials, 328 
shock-wave effect, 844 
water, 167 
Indium antimonide 
bolometers, 497 
detectors, 484-487 
reflection coefficient, 297 
semiconductor properties, 324 
Indium arsenide 
detector, 481, 482 
reflection coefficient, 297 
transmission, 324 
Indium phosphide 

reflection coefficient, 297 
transmission, 324 


Infrared systems design 
general procedures, 730 
gross analysis, 730 
mapping systems, 750 
optical systems, 732 
scanning dynamics, 734 
search system, 737 
sensitivity calculation, 731 
tracking system design, 743 
Inner loop gain distribution, 725 
Input 

control, 721 
displacement, 689 
noise equivalent, 731, 740, 753 
parabolic, 691 
ramp, 672, 687, 690, 694 
step, 672, 689, 693 
Insolation, see Radiation, solar 
Instantaneous fields of view, 739 
Insulating materials, 521 
Insulators, properties, 522 
Integral, convolution, 621 
Integral coolers, 529 
Integration time, 745 
Intercept, air-to-air, 744 
Interchangeable gratings, 771 
Interchangeable prisms, 721 
Interference filters, 286-294 
Interference fringes, 781 
Interferometer-spectrometer, 774 
Interferometers 
Fabry-Perot, 290, 778, 781 
spherical, 779 

Lummer-Gehrcke plate, 779 
Michelson, 777 
Rayleigh, 776 

spectral transmittance, 780 
Twyman-Green, 777 
Invariance, spatial, 621 
Invariant, Lagrangian, 433 
Inverse-Nyquist method, 677 
Inverse z-transforms, 698 
Inverting prism, 309 
Iridium, 806 
Iron, 806 
Iron oxide 
emittance, 362 
paint, 87 
Irradiance 
celestial bodies, 112 
definition, 784 
NBS standards, 40 
optical systems, 380 
visual magnitude, 114 
Irtran, 282, 294, 326 
Isoplanatic region, 621 
Isoplanatic systems, 614 

James-Weiss stability criterion, 672 
Jet propulsion systems, see Rocket and jet pro¬ 
pulsion systems 
Johnson noise, 469 


INDEX 


893 


Johnson solar spectral curve, 789 
Joint, thermal, conduction at, 786 
Joule-Thomson cooling systems, 523, 532 
Junction, cold, 550 
Junction transistor, shot noise, 586 
Jupiter, 794 


K-Monel, 804 
Kel-F, 326 

Kirchhoff diffraction integral, 635 
Kirchhoff diffraction theory, 615 
Kirchhoff’s law, 9, 27, 72, 795 
KRS-5 prisms, 309 
Krylon flat black paint, 362, 809 
Krypton, 179 


Lacquer, 359 

Lagrangian invariant, 373, 433 
Lambertian sources, 22 
Lambert’s law, 70, 795 
Laminar flow, 828 
Lamps, 40, 41, 49, 643 
Lamp black, 359 

Lateral chromatic aberration, 390-400 
Lateral magnification, 373 
Layers 
filters 

absentee, 288 
H&L, 288 

Herpin equivalent, 293 
single, reflection and transmission, 348 
Lead, 806 

Lead selenide detectors, 477, 478, 479 
Lead silicate glass, 318 
Lead sulfide detectors, 474, 475, 476 
Lead telluride 
detectors, 480, 836 
filter film, 295, 324 
Leidenfrost transfer, 531 
Length, conversion factors, 860 
Least squares design, 408 
Lenses 

achromats, 409, 282 
aplanatic, 380, 427, 434 
conical, 843 
Conrady G sums, 341 
corrector, 286 
design, 640 
field, 424, 732 
immersion, 427, 433 
materials, 282 
multielement, 282-284 
projection, 426 
relay, 426 
S.C.A, 282, 284 
single element, 282 
thin lens angular dispersion, 340 
Liaponov stability analysis, 704, 710 
Liaponov theorem, 711 
Light, incoherent, 621 


Light pipes, 427, 430, 431 
Linear systems 

analysis methods, 668 
definitions, 662 
feedback control system, 662 
general concepts, 614 
isoplanatic, 614 
operator, 614 

ramp inputs, 672, 687, 690, 694 
spatial domain, 614 
stationary (time invariant), 614 
superposition integral, 614 
type 0, 686 
type 1, 688 
type 2, 692 
transfer function, 664 

Linearization methods of optical design, 409 
Lines, spectral, 194, 291-293 
Linfoot quality factors, 637 
Liquid-feed coolers, 529 
Liquid-supply cooling systems, 533 
Liquid propellants, 64 
Liquid transfer coolers, 526 
Lithium fluoride 
Christiansen wavelengths, 296 
prisms, 309 

reflection coefficient, 298 
filter substrate, 294 
Littrow double-pass system, 773 
Littrow mirror, 771 
Littrow monochromator, 764 
Littrow-mounted spectrometer, 766 
Littrow prism, 308 
Loading resistance noise factor, 589 
Log modulus, 668, 682 
Long-wavepass interference filters, 290 
Longitudinal aberration, 381-386, 399 
Longitudinal chromatic aberration, 384-389, 
400 

Longitudinal magnification, 373 

Look, direction of, 97 

Loops 

closed, 708, 710, 722 
inner, gain distribution, 725 
motor driven, 688 
multiple, analysis, 725 
open, 707 

torqued, gearless, 692 
track, 692, 744, 746, 747 
Lorentz broadening, 178, 189, 191 
Lorentz line shape, 190, 194, 291 
Lorentz-Lorentz law, 839 
Loss tangent, 350 
Low-noise amplifiers, 597 
Low-noise cable, 611 
Low-noise transistors, 606 
Low-temperature liquids, 520 
Lucalox glass, 321 

Lummer-Gehrcke plate interferometers, 779 
Lumped-constant line, 293 
Luneberg apodization theorem, 621, 636 
Lyot filter, 299 
Lyot-Ohman filter, 298 


INDEX 


894 

M-Circle, 676 
Mach number 
effect on shock lens, 843 
free-stream, 848 
relation to Stanton number, 829 
shock-wave angle, 840 
Magnesia, 359 
Magnesium, 806 
Magnesium fluoride, 295, 836 
Magnesium oxide, 294, 296, 362 
Magnification, 373, 423 
Magnifying power, 373, 423 
Magnitude 
log, 678, 682, 720 
stellar, 107 

Maksutov optical systems, 285, 445 
Mangin mirror, 444 
Manganin, 807 

Mapping system, 734, 750-756 
Marginal rays, 399 
Margin 
gain, 664 
phase, 664 

Marine backgrounds, 166 
Mars, 794 
Masonry, 82 
Mass 

conversion factors, 866 
definition, 855 
units, table, 867 
Matrix, filter, 291 
Maximized D*, 467 
Maximum difference expression, 10 
Maxwell’s probability law, 585 
Mayer-Goody model, 194 
Mean-square noise voltage, 591 
Medium contrast film, 572 
Melting temperature, optical materials, 330 
Meniscus lenses, 282 
Meniscus corrector, 286 
Mercuric telluride detectors, 494 
Mercury (planet), 794 
Mercury 
arc, 49, 50 
Ge:Hg detectors, 492 
HgTe detectors, 494 
Merit factors, 637 
Merit functions, 408 
Meteoric flux, 793 
Meteoroid bombardment, 792 
Meteorology, 118 
Methane 

atmospheric absorption bands, 250 
atmospheric, 179, 187 
solid, temperature, 562 
Methyl group, 325 
Metric system prefixes, 857, 881 
Michelson interferometer, 777 
Microphonics, 461, 592 
Microscopic irregularities, 25 
Mie scattering, 205, 206 
Miller effect, 595, 608 
Minutes, radian equivalents, 865 


Mirrors 
chopper, 760 

commercially available, 285 
corner cube, 773 
Littrow, 771 
Mangin, 444, 452 
paraboloid, 773 
spinning, 506 
Mixing ratio, 182, 183 
Modulation noise, detectors, 469 
Molding compound, epoxy-type, 357 
Molecular emission of exhausts, 67 
Molecular groups, 325 
Molybdenum, 801 
Monel, 807 
Monochromators 
double, 765 
grating, 772 
Littrow, 764 
prism, 772, 766 
rapid-scan, 773 
single-pass, 765 
spectrometer use, 771 
Moon, 794 

Motor-driven loops, classical, 688 
Moving-reticle filters, 650 
Multicouple single-stage cooling systems, 555 
Multielement lens, 282, 284 
Multistage cooling systems, 555 
Multiple-detector array, 755 
Multiple-detector system, 753 
Multiple-loop analysis and synthesis, 725 
Mylar, 804 

n-type detectors, 488 
Nacreous clouds, 124 
Narrowband filters, 291 
National Bureau of Standards 
blackbody standards, 38, 44, 45 
carbon filament standards, 38 
comparison standards, 44 
glass, 321 

radiation standard, 39 
spectral irradiance standards, 40 
spectral radiance standards, 41 
total radiation standards, 38 
Negative feedback, 593 
Negative film, infrared, 571, 576, 580 
Negative transfer function, 629 
Neon, 179, 520 
Neptune, 794 
Nemst glower, 48 
Networks, 718, 724 
Newtonian optical equations, 372 
Newtonian telescope, 441 
Nichrome, 807 
Nickel, 807 
Night glow, 104 
Niobium, 805 
Nitrogen, 179, 520 
Nitrogen-hydrogen group, 325 
Nitrous oxide, atmospheric, 179, 187, 246 
Noctilucent clouds, 124 


INDEX 


895 


Noise 

amplifier sources, 584 
anode current flucuations, 585 
broadband figure, 609 
factors, 587 
detector, 588 
loading resistance, 589 
minimum, 602 
operating, 588 
overall, 590 
preamplifier, 589 
figures, 587, 606, 607, 609 
flicker current, 586 
generation-recombination, 470, 587 
generator, 589 
Johnson, 469 
low 

cable, 611 
design, 605 
modulation, 469 
Nyquist, 469 
output voltage, 464, 599 
peaked-channel, 591 
photon, 513 
planar transistors, 603 
power, ideal generator, 588 
semiconductors, junction transistor, 586 
shot, 470, 585, 586, 589, 591, 592, 597 
signal-to-noise ratio, 594, 740, 748 
source, 595 
spectral, 466 

spectrum, of detectors, 510 
temperature, 469, 588 
temperature-caused, detectors, 469 
thermal, 469, 584, 591 
total output, 591 
transistor, 597-602 
voltage, mean-square, 591 
Noise equivalent input, 731, 740, 753 
Noise equivalent power, 466, 501, 731, 753 
Noise equivalent resistance, 589 
Noise equivalent resistor, 590 
Noise equivalent voltage, 607 
Noise-limited systems, 751, 752 
Nonlinear systems 
analysis methods, 705 
bang-bang system, 715 
definitions, 703 

error plus error rate system, 714 
saturated, 713 

Nonoverlapping line approximation, 201 
Nonselective scattering, 206 
Normal emissivity, 795 
Normal emittance, 785 
Normalization methods, 761 
Nutation, 747 
Nuvistor, 590 
Nyquist methods 
inverse, 677 

linear system analysis, 668, 672 
sampled-data systems, 702 
Nyquist noise, detectors, 469 
Nyquist plot, design method, 718 


Ocean, 168-170 

Off-axis chromatic aberration, 385 
Off-axis response, 626 
One-third rule, 725 
Open-circuit impedance, 293 
Open-loop describing function, 707 
Operating noise factor, 588 
Operating temperatures, detectors, 463 
Optical Coating Laboratory filters, 299 
Optical component, 371 
Optical design 
automatic, 408 
electronic computers, 394 
primary aberration correction, 406 
general considerations, 406 
merit functions, 408 
techniques, 406 

Optical difference (wave front aberration), 616 
Optical element, 371 
Optical materials 

angular dispersion of thin lens, 340 
blacks 

emittance, 362 
optical properties, 356 
cooled 

description, 359 
transmittance, 361 
Conrady G sums, 341 
dielectric constants, 
dispersion, 328 
dispersion equations, 335 
dispersive power, 339 
equations, 347 
extinction coefficient, 351 
hardness, 333 

Herzberger dispersion equation, 337 
index of refraction, 328, 338 
loss tangent, 350 
radiation damage, 358 
softening or melting temperature, 330 
solubility, 333 
specific heat, 333 
surface coatings, 351 
thermal conductivity, 331 
thermal expansion, 332 
transmittance, 364 
Young’s modulus, 333 
Optical member, 371 
Optical parameters, 742 
Optical paths, shock wave, 843 
Optical systems 
See also Lenses 
aberration-free, 629 
afocal, 422 
anamorphic, 424 
aplanatic, 427 
arrays, 755 

cardinal points, 371-376 
cascaded, 626 
catadioptic objective 
Bouwers (Maksutov) system, 285, 445 
Maksutov, 285, 445 
Mangin mirror, 444 


896 


INDEX 


Schmidt system, 443 
transfer function, 624 
condensers, 424 
coverage, 732 
detector systems 

Abbe sine conditions, 428 
/7no., 429 

field lenses, 427, 429 
immersion lenses, 427, 433 
light pipes, 427, 430 
vignetting, 429 
diffraction-limited, 448 
erectors, 424 
image blur size, 450 
merit factors, 637 
multielement, 375 
periscopes, 424 
projection, 424 
radiometers, 762 
reflecting objectives 
baffling folded, 442 
Cassegrainian objective, 442 
folded reflector, 441 
Gregorian telescope, 442 
Newtonian telescope, 441 
optical collimator, 440 
parabolic reflector, 440 
spherical reflector, 437 
refracting objectives, 435, 454 
relay systems, 424 
transfer functions 
aberrations, 629 

annular and annulus apertures, 624 
apodization, 636 
cascaded systems, 626 
circular-aperture, 624, 630, 632 
coherent illumination, 626 
coma, 634 

computation, example, 634 
contrast ratio, 623 
defocusing, 630 
description, 621 
expressions, 624 

linear systems development, 614 
modulus, 623, 633 
negative, 629 
sampled-data systems, 700 
slit aperture, 630 
spherical aberration, 631 
Strehl criterion, 623 
typical reflecting system, 624 
windows, see Windows 
Optical thickness, 288, 294 
Optics 

See also Lenses; Optical systems 
Airy disc, 410 
apertures 
aberrations, 385 
annular and annulus, 616, 624 
circular, 616, 618, 624, 630, 654, 655 
cone, 377 

entrance, flux density, 730 
limit, 626 


rectangular, 619, 654, 655 
slit, 617, 619, 630 
stop, 371, 377-379 
aplanatic systems, 380 
back focal length, 376 
baffles, 379 

cardinal points, 371-376 

combination of two elements, 376 

definitions, 410 

depth of field, 380 

depth of focus, 380 

diffraction theory 

annular and annulus aperture, 616, 624 
circular aperture, 616, 618 
complex amplitude, 615 
defocusing, 618 
illuminance, 615 
Kirchhoff theory, 615 
point spread function, 615 
pupil function, 616 
rectangular and slit aperture, 619 
Straubel pupil function, 618 
Strehl criterion, 621 
variable pupil functions, 620 
effective focal length, 376 
energy distribution, 410 
entrance pupil, 379 
equations, 372-375 
fl no., 379 
field stop, 378 
focal points, 371 

frequency response see Response, frequency 

front-end description, 747 

geometry, 371 

glare stops, 379 

image irradiance, 380 

image spot size, 410 

Lagrangian invariant, 373 

magnification, 323 

numerical aperture, 379 

paraxial rays, 382 

paraxial ray trace, 377 

principal points, 371-375 

principal ray, 376-377 

pupils, 377 

ray tracing, 375 

Rayleigh criterion, 412 

relative aperture, 379 

resolution, 410 

sign convention, 371 

speed, 379 

Snell’s law, 371 

symbols, 371 

vignetting, 378 

Optics Technology, Inc., filters, 299 
Optimum bias of detectors, 502 
Optimum noise figure, 607 
Ordnance materials, 74 
Oscillations (high-reflectance zone), 290 
Output 

control, first-derivative method, 721 
noise voltage, 599 
spatio-temporal, filters, 649 


INDEX 


897 


Overshoot, linear system, 662 
Oxygen, 179, 520 
Ozone, atmospheric 
distribution, 185 
emission peak, 96 

high-altitude maximum concentration, 187 
percentage, 179 
relation to air glow, 105 
seasonal variation, 186 
ultraviolet radiation, 185 

p-type detectors, 489 
Paint 

See also Enamel 
aluminum, 78, 809 
Centerlite white, 91 
Codit silver, 90 
Flex-o-lite beaded, 92 
gold, 79 

green masonry, 82 
Krylon black, 809 
No. 13, 86 
red iron oxide, 87 
thermal radiation properties, 807 
Palladium, 809 
Palmer scan, 736 
Parabolic input, 691 
Parabolic reflector, 440, 452 
Paraboloid, optical properties, 440 
Paraboloid mirror, 773 
Parameters 
cooling, 742 
optical, 742 
sensitivity, 740 
signal processing, 742 
system block diagram, 742 
systems, 730 
windows, 836 
Paraxial focal plane, 386 
Paraxial focus, 634 
Paraxial marginal ray, 402 
Paraxial principal ray, 402 
Paraxial rays, 377, 382, 390, 399 
Parseval’s theorem, 647 
Partial dispersion, 403 
Partial reflectance, 28 
Particle-size distribution curves, 188 
Paschen-Runge grating mounting, 312 
Passband filters, 287 
Path length, 116 
Paths, atmospheric, 263-266 
Peaked-channel noise, 591 
Peltier coefficient, 548 
Peltier coolers, 523, 546 
Peltier couple, performance, 549 
Pentode, 586, 592, 595 
Periscopes, 424 

Petzval contribution to aberration, 400 
Petzval curvature, 385, 439, 440 
Petzval surface, 384, 400, 439-440 
Pfund grating spectrometers, 767 
Pfund radiometer, 762 


Phase angles, 736 
Phase difference, filters, 288 
Phase lag network, 718 
Phase margin, linear systems, 664 
Phase-plane 
analysis, 709 
definition, 704 
response time, 710 
Phase plots (nonlinear systems), 720 
Phase portrait, 709, 713 
Photoconductive detectors, 458 
Photodetectors, 458 
Photoelectromagnetic detectors, 458 
Photoemissive detectors, 458 
Photographic film, see Films and plates 
Photometer, double beam, 772 
Photovoltaic detectors, 458 
Photon emission rates, 106 
Photon noise limitation, detectors, 513 
Phthalocyanide, sublimated, 306 
Physical constants, values, 855, 858 
Pitch, coal tar, 84 
Pitot pressure, 830 
Planar transistors, 603 
Planck blackbody function, 795 
Planck’s law, 9, 20 
Plane of best focus, 638 
Plane of the ecliptic, 785 
Planetary albedo, 814 
Planetary inclination, 785 
Planetary radiation 
definition, 812 
incident, 819 
thermal, geometry, 817 
Planets 

effective temperature, 112 
visual magnitudes, 112 
Plastics 

absorption wavelengths, 326 
filter materials, 306-307 
Kel-F, 326 
Platinum, 359 
Plexiglass, 325 
Pluto, 794 
Points 

principal, optics, 371, 375 
singular 

linear systems, 672 
nonlinear systems, 704 
compensation methods, 723 
Point sources, 51, 638 
Point spread function, 615 
Poisson distribution (noise), 585 
Polar coordinates, 647 
Polarization, filters, 286, 298 
Polarizers, 298 
Polyethylene, 325 
Porro prism, 309 
Potassium bromide 
Christiansen wavelength, 296 
prisms, 309 

reflection coefficient, 298 
Potassium chloride, 298 


898 


INDEX 


Potassium iodide, 298 
Power 

conversion factors, 879 
definition, 857 
dispersive, 339 
effective, radiometers, 763 
emissive, earth, 790 
gain, available, 588 
maximum transfer, 293 
resolving, 618 
responsivity, 761 
supply, 597 

Power-density spectrum, 726 
Prandtl number, 829 
Preamplifiers 
available power gain, 588 
FET design, 608 

high-frequency compensation, 590 
low-noise design, 605 
mean-square noise voltage, 591 
noise factor, 589 
Pressure 

conversion factors, 872 
definition, 856 

generator, Barnes nitrogen, 530 
regulator, 533 
total, atmospheric, 189 
Principal point, optics, 371, 375 
Principal rays, 399 
Prisms 
Abbe, 308 
Amici, 308 
characteristics, 307 
constant-dispersing, 308 
deviating, 308 
dispersing, 308 
Dove, 308 
fore, 766 

interchangeable, 771 
inverting, 309 
Littrow, 308 
materials, 309 
monochromator, 766, 772 
Porro, 309 

position in infrared system, 736 
roof, 308 
rotation, 734 

spectroradiometric uses, 763 
total reflection, 309 
triple mirror, 309 
Wadsworth, 308 
Probability 
of detection, 740 
gaussian, 740 
Probes, deep space, 811 
Projected solid angle, 24 
Projection lenses, 426 
Projection systems, 424 
Propellant fuels, 62, 67 
Proportionality condition, 614 
Pulse response, detectors, 505 
Pulse-position system, 746 


Pupils, entrance and exit, 377-379, 616-621, 
624, 636, 733 

Pyromark standard black, 362 

Quadrature method, gaussian, 631-633 
Quantum efficiency, detective, 468 
Quantum rates in blackbody radiation, 10 
Quarter-wave optical thickness, 288 
Quarter-wave stack, 288 
Quartz 

Christiansen wavelengths, 296 
crystal, prisms, 309 
filter uses, substrate, 294 
glass, fused, 317 
iodine lamps, 40 

thermal radiation properties, 809 
windows, 39 

Quasi-random model, 196-201 

Radiance 
absorbed, 27 
background 
winter day, 164 
winter night, 165 
blackbody, 96 
brick wall, 146 
concrete, 146-156, 164 
definition, 784 
diurnal variations, 144, 149 
grass, 146, 157, 164 
ground, 143 
incident, 23, 27 
NBS standards, 41 
ocean, 169 
opaque body, 70 
reflected, 24, 27 
sky, 96-99, 143, 164 
snow, 146, 163-164 
terrain, see Terrain 
urban area, 143 
Radiancy, 784 

Radians, degree equivalents, 865 
Radiant emissivity, see Emissivity 
Radiant emittance, 23 
Radiant energy reference level, 758 
Radiant heat transfer, 787 
Radiant intensity, 24 
Radiation 
albedo, 790,812 
atmosphere, 737 

atmospheric temperature effects, 97 
background, 

discrimination, 737 
high-speed flight, 826 
minimization, 737, 739 
blackbody, 10 
boundary-layer effects, 846 
dosages, 360 
earth, 785 

equations and constants, 28 
geometry, 22 
heat transfer, 828 
heated air, 851 


INDEX 


899 


laws, 9-10 

NBS standards, 38 

planetary, 112, 812, 819 

ratio of visible to infrared, 570 

reference level, 759 

rocket and jet propulsion systems, 68 

reflected solar, 785, 790, 812, 814, 816 

shock wave effects, 841 

solar insolation, 96, 785, 812, 813 

space, 820, 821 

stellar, 107, 110 

target, 737 

thermal, 

planetary, 817 

properties of selected materials, 804 
temperature of atmosphere, 97 
total, of targets, 71 
transducer, 458 
ultraviolet, in space, 792 
ultraviolet, effect on atmospheric ozone, 185 
windows, 827, 833 
Radiators, distributed, 23 
Radiometers 

alternating current, 760 
basic design, 758 
Cassegrain, 762 
commercially available, 761 
direct current, 759 
effective power, 763 
essential components, 758 
normalization to the peak, 763 
Pfund, 762 

principal characteristics, 759 
spectral bandwidth, 763 
spectroradiometers, 763 
Radiometric quantities, 4, 761, 784 
Radiometric relations, 28 
Radius of curvature, 840 
Random Elsasser model, 196-201 
Random process Fourier transfer, 649 
Range 

calculation, 747 
conversion chart, 863 
detection, 739 

Rapid-scan monochromator, 773 

Rapid-scan spectrometer, 773 

Raster scan, 736 

Rate error linear systems, 663 

Rays 

limitation by stops and aperture, 377 
marginal, 399 
marginal paraxial, 402 
paraxial, 382, 399, 402 
principal, 376-377, 399 
principal paraxial, 402 
skew trace, 396 
Ray tracing 

aberrations, 388 

Coddington’s equations, 392, 395 
desk calculator, procedures, 390 
electronic computers, 394 
graphical, 398, 431 
paraxial, 377, 390 


precision, 388 
series of elements, 375 
Snell’s law, 388 
transfer function, 634 
Rayleigh criterion, 412, 627 
Rayleigh interferometers, 776 
Rayleigh-Jeans law, 10, 20 
Rayleigh scattering, 204, 208 
Receiver-collector area, 224 
Reciprocal dispersion, 403 
Reciprocity, films and plates, 574 
Reciprocity law, 27 
Reciprocity-law failure, 574 
Reciprocity theorem, Helmholtz, 24, 27 
Recovery factor, 829 
Rectangular aperture, 619 
Rectangular aperture filter, 654 
Reentry heating, 69 
Reference blackbodies, 761 
Reference radiation level, 759 
Reflectance 

aluminum foil, 76 
asphalt, 83 
coaltar pitch, 84 
coating materials, 80, 354 
definition, 784 
diffuse, 70 
directional, 23, 26 
enamels, 80 
fabrics, 76 

high, zone filters, 280 
masonry, 82 
measured values, 74 
nonequlibrium condition, 27 
optical surface coatings, 351 
ordnance materials, 74 
paints, 80 
partial, 25, 28 
rubber, natural, 75 
specular, 70 
steel, 76 
targets, 72 

terrain features, 75, 85 
total, 25, 28 

Reflected solar radiation, see Albedo 
Reflection, see also Reflectance 
coefficient, 293, 297 
distribution function, 24 
filters, selective, 286, 297 
internal 

immersion lenses, 433 
light pipes, 431 
loss 

different incidence angles, 350 
tangent, 350 
prism, total, 309 
single layer, 348 
single surface, 347 
stop band, 290 
water surface, 167 
Reflectivity 

angular dependence, 793 
definition, 784 


900 


INDEX 


directional, 88-93, 793 
interference filters, 292 
maximum, filters, 288 
partially transparent bodies, 349 
reciprocity relation, 796 
seawater, 166 
selected materials, 796 
terrain, 142 
water, 167 

wave slope, effect, 168 
Reflectors, 437-440, 450 
Refracting systems, 454 
Refraction, 286, 298, 839 
Refractive index, see Index of refraction 
Relative noise temperature, 588 
Relative stability (linear systems), 664 
Relative visibility curve, films, 573 
Relaxation methods of optical design, 409 
Relay lenses, 426 
multielement, 284 
Relay systems, 424 
Replica mirrors, 355-358 
Representations-of-the-system method, 702 
Residual aberrations, 386, 407 
Resistance 
detectors, 462 

equivalent noise, 589, 593 * 

Resistors, 590 
Resolution 

aberration limits, 414 
criteria for point sources 
Sparrow resolution, 628 
Rayleigh resolution, 627 
energy distribution, 410 
limit, aerodynamic effects, 850 
loss, due to turbulent boundary layer, 849 
sine-wave, 628 

spectral, in monochromators, 765 
spurious, 629, 637 
theoretical limit, 844 
Resolving power, 618 
Response 

amplifiers, high-frequency, 595 

apodization, 636 

frequency 

calculations of design, 680 
computer calculation, 640 
detectors, 504 
FET amplifiers, 609 
optical, 410 
track loop, 746 
impulse, 614 
off-axis, 626 
optical, 643 
pulse, of detectors, 505 
spectral, of detectors, 508 
time, from phase plane, 710 
transient, analysis, 667, 675, 702 
Responsive area, detectors, 462 
Responsive element, detector, 458 
Responsive quantum efficiency, detectors, 468 
Responsivity 
blackbody, detectors, 464 


power, 761 

spectral, of detectors, 464 
Reticles 

circular sectored, 653 
filters, circular sectored, 655 
moving, 651 

infinite checkerboard, 654, 655 
infinite parallel spoke, 654, 685 
square wave, checkerboard, 651 
wagon-wheel, 653 
Reynolds analogy, 829 
Reynolds numbers, 828 
Rhodium, 351, 809 
Ribbon filament lamp, 58, 643 
Rim-ray curve, 386-387 
Rise time (linear systems), 662 
RMS noise voltage, detectors, 464 
RMS signal voltage, detectors, 463 
Rocket and jet propulsion systems 
afterburning heat release, 60 
exhaust 

composition, 62 
emission bands, 64 
equilibrium constant, 66 
first-period length, 61 
flow field, 61 
in vacuum, 62 
jet structure, 60 
major emission bands, 67 
molecular emission, 67 
nozzles, 61, 62 
particles, 67 

stagnation temperature, 61 
temperatures, 59 
undisturbed cone, 62 
velocity, 59 
exit heating, 68 
heat flow, 68 

high-energy solid fuels, 67 
liquid propellants, 64 
mass flow, 59 
propellants, 62, 67 
radiation processes, 68 
reentry heating, 69 
shock-wave formation, 61 
temperature, 68 
thrust, 59 

Rock salt prisms, 309 
Rokide, 809 

Roof prism (Amici), 308 
Root locus methods, 
linear system, 668, 683 
sampled-data systems, 703 
systems design, 722 
Rosette scan, 736 
Rotating prism, 734 
Rotating wedges, 735 
Rotation, rates of, 734, 736 
Routh’s stability test, 669 
Rowland circle focal curve, 311 
Rowland grating mounting, 312 
Rubber, 75 
Rubidium iodide, 296 


INDEX 


901 


s-plane 

contour (linear systems), 673 
transfer functions, 700 
Sagittal coma, 382, 438-439 
Sagittal focus, 384 

Sample function (Fourier transforms), 649 
Sampled-data systems 
analysis methods, 701 
absolute stability, 700-702 
definitions, 695 
design, 719 

transfer functions, 700 
types, 703 
Sampler, 696 
Sampling, 695 
Sampling theorem, 696 
Sapphire, 294 
Satellites, 812 
Saturation 
detectors, 834 
systems with, 713 
Saturn, 794 
Sawyer glass, 318 
Scan patterns, 736 
Scanning, 734, 747 
Scanning-aperture space filters, 650 
Scanning-field space filters, 652 
Scattering, 96, 286 
Scattering angle, 117 
Scattering coefficients, 204-211 
Schmidt system, 443, 453 
Schur-Cohn stability criterion, 702 
Scintillation 

See also Stars, stellar scintillation 
atmosphere, 210 

turbulent boundary-layer effect, 849 
Sea, see Ocean 
Sea water, 166 

Search system design analysis, 737 
Seasonal variations atmosphere, 186 
Secondary maxima, diffraction pattern, 618 
Seconds, radian equivalents, 865 
Seebeck coefficient, 544, 548 
Seidel aberrations, 382-390, 630 
Selective absorption, 286 
Selective radiators, 23 
Selective reflection filters, 297 
Selective refraction filters, 298 
Selenium glass, 321 
Semiconductors, 306-307, 324 
Sensitivity 

analysis, 731, 744 
calculation, equations for, 731 
contours, detections, 510 
design analysis, 738 
front-end description, 747 
mapping, 731 
optical systems, 732 
parameters in system design, 740 
spectral, films, 574, 578 
target-detection, 731 
tracking systems, 744 
Sensitometric characteristics, films, 573 


Servo analysis, 745, 747 
Servo-bandwidth-limited system, 750 
Servo Corp. of America 
catadioptric systems, 285 
lenses, 282, 284 
Servocon lenses, 282 
Setting time (linear systems), 663 
Seya-Namioka grating mounting, 312 
Shape factor, 435 
Sheet iron, 806 

Shift, angle, interference filters, 290 
Shimmer, see Atmosphere, scintillation; Stars, 
stellar scintillation 
Shock effect on detectors, 461 
Shock formation theory, 61 
Shock waves 
aberrations, 842 
angle, 840 
effects, 839 
on focal length, 844 
on index of refraction, 844 
on infrared radiation, 841 
on resolution, 844 
on stagnation temperature, 83 
hemispherical, 846 
supersonic, 61 

Short circuit impedance, 293 

Short-wavepass interference filters, 290 

Shot noise, 470, 585, 586, 589, 591, 592, 597 

Sicon black, 47, 362 

Signal flow diagrams, 664 

Signal flow rules, 665 

Signal generator, 589 

Signal processing, 742 

Signal-to-noise ratio, 594, 740, 748 

Signal voltage, RMS, detectors, 463 

Silicate glass, 321 

Silicon 

detectors, 473 

effect of neutron bombardment on absorption 
coefficient, 358 
filter film, 295 
filter substrate, 294 
lenses, 282 

neutron bombardment, 358 
solar cell, 810 
transistor, low noise, 601 
transmission as semiconductor, 324 
Silicon-oxygen group, 325 
Silicon carbide, 359 
Silicon:gold detectors, 496 
Silicon monoxide 
emissivity, 359 
filter film, 295 
filter mirror coating, 352 
reflectance, 354 
transmittance, 356 
Silicon:zinc detectors, 495 
Silver, 351, 810 
Silver chloride, 309 
Silver paint, Codit, 90 
Simple compensation networks, 718 
Simple speed controllers, 686 


902 


INDEX 


Simpson’s rule, 633 
Simulator, blackbody, 32 
Sine condition, 282, 389, 428 
Sine waves 

modulation, 214 
resolution, 628 
spatial, 643 
target, 638 

Single cell detector cooling systems, 557 
Single-element lenses, 282 
Single-element objectives, 434 
Single-pass monochromator, 765 
Single surface reflection and transmission, 347 
Single-pulse system, 748 
Single-stage multicouple cooling systems, 555 
Single-stage thermoelectric cooling 
systems, 525, 554 
Singular points, 672, 704, 723 
Sintered metal filters, 533 
Skew ray trace, 396 
Sky, see Backgrounds, sky 
Slant paths, atmospheric, 261-266 
Slant range, 115 
Slide rules, blackbody, 11 
Slit 

aperture, 617-619, 630 
entrance, monochromator, 766 
Slope (filters), 287 
Smoke, 203, 356 
Snells’ law, 371, 388, 398 
Snow, 142, 146, 163 
Sodium aluminum fluoride, 356 
Sodium bromide, 296 
Sodium chloride, 295, 296, 298 
Sodium fluoride, 298 
Sodium iodide, 296 
Solar absorptance, 792, 800 
Solar activity, 101 
Solar heat flux, 811 
Solar radiation, see Radiation, solar 
Solar spectrum measurements, 227-229 
Solar system, 793 
Solder, 810 
Solid light pipes, 431 
Solubility of optical materials, 333 
Solution of aberration problems, 400 
Soot, 359 
Sources 
artificial, 32 
area, 223, 225 
blackbody (2400°K), 47 
blackbody simulator, 32 
carbon arc, 49 
cavity, 32, 51 
detector uses, 501 
field, 46 
Globar, 48 
laboratory, 46 
Lambertian, 22 
lamps, 49, 50 
low-temperature, 47 
mercury arc, 49, 50 
NBS, 38 


Nemst glower, 48 
Sicon-black enamel, 47 
Welsbach mantle, 48 
Zapon paint, 48 

Space and space frequency differentiation, 647 
Space charge-limited diodes, 586 
Space environment, 792 
cooling systems, 561 
Space-frequency shifting, 646 
Space radiation, 820 
Space scaling, 646 
Space shifting, 646 
Space technology terminology, 785 
Spacecraft 

See also Rocket and jet propulsion systems 

components, temperature range, 822 

deep space probes, 811 

passive control, 822 

testing, 823 

thermal coatings, 787 

thermal design, 821 

Sparrow method of calculating emissivity, 37 
Sparrow resolution criteria, 628 
Spatial domain, 623 
Spatial filter analysis, 739 
Spatial filtering, 737 
Spatial frequencies, 626 
Spatial frequency domain, 628 
Spatial frequency filtering, 646 
Spatial invariance, 621 
Spatio-temporal filter, 649 
Specific heat of optical materials, 333 
Spectral bands and lines 
absorption, 238 
admittance (filters), 293 
carbon dioxide, 238, 239, 240, 242, 243 
carbon monoxide, atmospheric, 249 
equally intense distribution, 194 
exponential distribution of line intensities, 
194 

infinite distributed, 293 
infinite lumped-constant, 293 
intensity, 194 
Lorentz shape, 291 
methane, atmospheric, 250 
models, 192 
narrowness, 291 
nitrous oxide, atmospheric, 246 
terminated by z«, 293 
transmission-line theory, filters, 293 
water vapor (atmospheric), 244 
Spectral bandwidth, 763 
Spectral classes of stars, 107 
Spectral curve, Johnson solar, 789 
Spectral D *, 467 
Spectral D**, 467 
Spectral detectivity, 466 
Spectral distribution, 761 
Spectral emissive power, earth, 790 
Spectral emittance, 785 
Spectral irradiance, see Irradiance 
Spectral noise equivalent power, 466 
Spectral purity, 765 


INDEX 


903 


Spectral radiance, see Radiance 
Spectral reflectance, see Reflectance 
Spectral response detectors, 464, 508 
Spectral sensitivity, 574 
Spectrometers 
commercially available, 767 
double-pass, 771 
flame temperature, 772 
grating, 766 
in-line, 774 
interferometer, 774 
Littrow mounting, 766 
Pfund grating, 767 

prism-grating double monochromator, 771 
rapid-scan, 773 
Spectrophotometers 
direct-ratio, 776 
double-beam optical wall, 775 
Spectroradiometers, 763 
Spectroscopic energy, conversion factors, 878 
Spectroscopic films, Kodak, 577, 572-573, 576- 
579 

Spectrum analyzer, 726 
Speed 

aperture, 379, 856, 869 
conversion factors, 856 
definition, 869 

Speed controllers, simple control systems, 686 
Spherical aberration, 382-388, 399-400, 414, 
435-439, 446-447, 629-633 
Spherical Fabry-Perot interferometer, 779 
Spherical mirror, 437 
Spherical reflector optical system, 437-439 
Spherochromatism, 386 
Spike, interference filters, 292 
Spinning-mirror technique, 506 
Spiral scan, 736 
Spot diagram, 389 
Spread function, 617 
Spurious resolution, 629, 637 
Square-band interference filter, 291 
Square coefficient (absorptance), 196 
Square-wave reticle filters, 651 
Square root region, 191 
Stability 
absolute 

analysis method, 669, 672, 674, 681, 685, 
700 

linear systems, 664, 700-702 
nonlinear systems, 702 
sampled-data systems, 700 
asymptotic, 704 
criteria 

James-Weiss, 672 
Schur-Cohn, 702 
describing function analysis, 708 
global, 704 
Liaponov, 704, 710 
relative (linear systems), 664 
test, Routh’s, 669 
transistors, 598 
Stabilization 
bias, 598 


requirements of mapping systems, 755 
track, 750 

Stack, quarter-wave, 288 
Stagnation temperature, 61, 831 
Stainless steel, 76, 801, 810 
Standards 

atmospheric temperature profiles, 177 
carbon filament, 30 
length, 855 

NBS emissivity comparsion, 44 
NBS radiation, 38 
NBS spectral irradiance, 40 
NBS spectral radiance, 41 
noise temperature, 588 
Pyromark black, 362 
Stanton number, 828 
Stars 

apparent galactic concentration, 109 

background radiation, 107 

concentration by spectral class, 109 

effective temperature, 112 

numbers, by magnitude, 107 

spectral classes, percentages, 107 

spectral classification, 107 

spectral distribution of stellar radiation, 110 

spectral irradiance by visual magnitude, 114 

stellar scintillation 

aperture size, effect of, 218, 221 
crossover frequency, 215 
effects of upper air winds, 216 
zenith distance, 218, 222 
surface temperature, 107 
visual magnitudes, 112 
Stationary systems, 614 
Statistical model, 194-199 
Steady-state behavior, 674, 681, 685 
Steel, stainless, 76, 359, 810 
Steepest descent design, optics, 408 
Stefan-Boltzmann constant, 39 
Stefan-Boltzmann law, 10 
Stellite, 351 

Step inputs, 672, 689, 693 
Steradiancy, 784 
Stirling cycle, 540 
Stop band, 287, 290 
Stop shift theory, 404, 440 
Stratosphere 

aerosol content, 141 
clouds, 124 
dry, 181 

water vapor, 182, 183 
wet, 181 

Straubel pupil function, 618 
Strehl criterion, 621, 633, 636 
Strip filament lamps, 50 

Strong-line approximation of absorptance, 191, 
200 

Sub-auroral belts, 101 
Substrates, filter, 287, 294 
Sun, 794 

See also Insolation; headings beginning Solar 
Superposition, linear systems, 614 
Supersonic shock waves, 61 


904 


INDEX 


Surfaces 

aplanatic, 427 
aspheric, 385, 402 
coatings, 352 
irregularities, 25 
Petzval, 384, 400, 439-440 
targets, see Targets 
thermopile, 40 
Symmetrical principle, 406 
Symmetry, axis of, 844 
Synchronous detection, 610 
Systems, see system names 

T-12 (optical material), 326 
Tabor, 810 
Tangent, loss, 358 
Tantalum, 803, 810 
Target presence generation, 750 
Target-detection sensitivity, 731 
Targets, 58 

See also Rocket and jet propulsion systems 
blast furnaces, 69 
concrete, 146 
contrast, 71 

directional reflectivity, 88-93, 793 
greybody, 70 

passive, temperatures of, 72 
perfectly diffusing surface, 70 
power plants, 69 
radiance, 70 

radiation design consideration, 737 
reflectance, 70 
supersonic shockwaves, 61 
thermal emission, 70 
total radiation, 71 
vehicle exhausts, 69 
Telescope 

astronomical, 422 
Cassegrain, 442 
Galilian, 422 
Gregorian, 442 
Newtonian, 441 
terrestrial, 422 
Tellurium detectors, 483 
Temperature 

ambient, effects on radiance of 
snow, 163 

conversion factors, 873 
definition, 856 
detectors 

background, 462 
cycling tests, 461 
noise, effect, 469 
operating, 463 

ranges of detector cooling system, 525 
storage requirements, 461 
effects on atmospheric scintillation, 223 
effects on semiconductors, 324 
gradients, 833 
high-speed flight 
rate of increase, 832 
recovery, 832 
wall, 829 


local static, 831 
low-, liquids, 520 
lowest, for liquid gases, 521 
maximum (recovery), 832 
methane, solid, 562 
noise, 588 
optical materials 
melting, 330 
specific heat, 333 
thermal conductivity, 331 
thermal expansion, 332 
profiles, 177-180 

ranges, of spacecraft components, 822 
sea-surface, distribution, 169 
sea boundary layer structure, 170 
stagnation, 61, 831 
stellar, 107 
terrain, 142 
window, 833, 837 
Terrain 
airfield, 144 
bare ground, 142-143 
city, 144 

concrete, see Concrete 

diurnal variations in radiation, 143 

dry sand, 144 

emissivity, 75, 142 

forest, 144 

grass, 144, 146, 157, 164 
moist sand, 144 
ocean, 142 

radiance, spectral, 142 
radiation of selected backgrounds, 144 
reflectance, 75, 85 
reflectivity, 142 
snow, 142, 146, 163, 164 
spectra, 143 
temperature, 142 
urban area, 143 
Terrestrial telescope, 422 
Thallium 
bromo-iodide, 309 
bromide, 298 
chloride, 297 
iodide, 296, 298 
Theoretical limit 
detectivity, 512 
resolution, 844 
Thermal 

coatings for spacecraft, 787 
conductivity, 331 
contact conductance, 786 
detectors, 458, 459 
emission, 70, 96 

expansion of optical materials, 332 
joint conduction, 786 
loads, 827 
measurements, 799 
noise, 469, 585 
radiation, see Radiation 
Thermistors, 458, 498 
Thermocouples, 458, 499 
Thermoelectric cooling systems, 523, 544 


INDEX 


905 


Thermopiles, 40, 458 

Thermopneumatic detectors, 458-459, 500 

Thermovoltaic detectors, 459 

Thevenin equivalent, 590 

Thick lens, 375 

Thickness, optical, 288, 294 

Thin lens, 340, 372, 375, 404 

Thin element optical systems, 436 

Third-order aberrations, 382-390, 404, 414 

Thomson relations, 548 

Thorium fluoride, 295 

Three-element achromat, 284 

Threshold, operating, 740 

Thrust, 59 

Time 

behavior (linear systems), 672 
constant, 662 

detectors, 465, 504, 738 
conversion factors, 868 
definitions, 856 
delay, 662 

dwell, 730, 734, 739, 752 
integration, 745 
response, 663 
from phase-plane, 710 
rise, 662 
to peak, 663 

Time-invariant systems, 614 
Tin, 811 

Titanium, 351, 810 
Titanium dioxide, 295 
Titanox-RA, 810 
Torque 

conversion factors, 871 
definition, 856 
Total absorptance, 28 
Total emissivity, 28 
Total emittance, 785 
Total power, 437 
Total radiation, 71, 821 
Total reflection prism 309 
Track accuracy, 746 
Track stabilization, 750 
Track-field size, 745 
Track-loop 
definition, 744 
servo analysis, 747 
frequency response, 746 
Track-while-scan system, 745 
Tracking system, 744 

many tracking loops system, 692 
Transconductance, dynamic, 586 
Transducer, 458 

Transfer, heat, see Heat transfer 
Transfer efficiency, cooling systems, 531 
Transfer functions 

See also Optical systems, transfer function 
closed loop, 722 
linear systems, 664 
s-plane, 700 

sampled-data systems, 700 
Transform relationships, 696 
Transformation analysis, 702 


Transformers 
broadband, 604 
coupling, 604 
design, 604 

Transient behavior, 674, 681, 685 
Transient response analysis 
linear systems, 667, 675 
sampled-data systems, 702 
Transistors, 586, 597-602, 604, 606 
Transmission 
atmospheric, 737 
efficiency, light pipes, 431 
Lyot filter, 299 
semiconductors, 307 
through the atmosphere, 252 
through clouds, 258 
window, at elevated temperatures, 833 
Transmission-line theory, 293 
Transmission region of optical materials, 326 
Transmissivity 

calculation of atmospheric slant paths, 266 
partially transparent bodies, 349 
sea water, 166 
Transmittance 

antireflection coatings, 356 
cooled optical materials, 359-361 
definition, 784 

filters (photographic), 574, 580 
glass 

calcium aluminate, 323 
Corning No. 18655, 318 
Germanate, 322 
high-silica, 318 
interferometers, 780 
optical materials, 364 
peak, 287 
polyethylene, 325 
semiconductor materials, 324 
single layer, 348 
single surface, 347 
substrate, 287 

Transverse aberrations, 381, 385, 390, 399 

Trap, cold, 533 

Triode, 589, 592 

Triple mirror prism, 309 

Triple point, 520 

Tropopause, 122-123 

Tubes, vacuum, 592-595 

Tungsten, 803, 811 

Tungsten filament lamp, 40, 49, 643 

Tungsten oxide, 356 

Turbine expander cooling systems, 544, 545 
Turbulent flow 

aerodynamic effects, 848 
recovery factor, 829 
Stanton number, 831 
transition from laminar flow, 828 
Twilight airglow, 106 
Twinkle, see Atmosphere, scintillation 
Twyman-Green interferometer, 777 
Type 0 linear system, 686, 703 
Type 1 linear system, 688, 703 
Type 2 linear system, 692, 703 


906 


INDEX 


Ultraviolet absorptance, 792 
Ultraviolet radiation in space, 792 
Upper air winds, 216 
Uranus, 794 

V number, Abbe, 403, 409 

V value, 435 

Vacuum environment specifications, detectors, 
461 

Vacuum tubes, 592, 593, 595 
amplifiers, 592 
Velocity 

conversion factors, 869 
definition, 856 
Venus, 794 

Vertical blackbody, 45 
Very-high-impedance amplifiers, 596 
Vibration requirements, detectors, 461 
View angles, 115 
Visibility-, relative, 573 
Vignetting, 378, 429 
Vinyl, 76 
Voltage 

output noise, 599 
RMS signal, detectors, 463, 464 
Volume 

conversion factors, 864 
definition, 855 
Vycor glass, 294, 317 

Wadsworth grating mounting, 312 
Wadsworth prism, 308 
Water, 167, 362 
sea, 167 

Water vapor, 96, 178, 181, 244 
Wave, symmetrical triangle, 651 
Wave number, 288 
Wave slope, 168 
Wave-front aberration, 616 
Wavelength 
detectors 
cutoff, 468 
peak, 467 
filters, 287 
operation, 738 

Weak-line approximation, 197-198 
Wedges, rotating, 735 

Weighting function (linear systems), 663, 672 
Welsbach mantle, 48 


White enamel, 359 
Wind, 216, 223 
Wien law, 10, 20 
Wiener spectrum, 649, 651 
Windows 
aberrations, 826 
detectors, 359 
ease of fabrication, 827 
emissivity, 827 
entrance and exit, 378 
fluorite, 39 
heating of, 827 
hot-, problem alleviation, 836 
materials 

characteristics, 833 
emissivity, 835 
magnesium fluoride, 836 
parameters, 836 
quartz, 39 

radiating, effects, 833 
radiation, 827 

at elevated temperature, 833 
requirements, 826 
temperature rise, 837 

transmission at elevated temperature, 833 
Wool, 76 
Work 

conversion factors, 875 
definition, 857 

Xenon, 179 

Young’s modulus, 333 

z-plane, transfer functions, 700 

z-transform, 698 

Zapon paint, 48 

Zenith, 96, 99, 105, 115, 143 

Zero-order circuits, 696 

Zinc 

Ge-Si:Zn detectors, 495 
Ge:Zn detectors, 493, 495 
Zinc sulfide, 295, 297, 356 
Zinc oxide paint, 86 
Zirconia, 359 
Zirconium, 51 
Zirconium dioxide, 295 
Zonal aberrations, 386 
Zone, auroral, 101 
Zone, high-reflectance, 288-290 


*U. S. GOVERNMENT PRINTING OFFICE : 1966 O - 236-960 

































































































